Radical factorization in finitary ideal systems

Abstract In this article, we investigate the concept of radical factorization with respect to finitary ideal systems of cancellative monoids. We present new characterizations for r-almost Dedekind r-SP-monoids and provide specific descriptions of t-almost Dedekind t-SP-monoids and w-SP-monoids. We show that a monoid is a w-SP-monoid if and only if the radical of every nontrivial principal ideal is t-invertible. We characterize when the monoid ring is a w-SP-domain and describe when the *-Nagata ring is an SP-domain for a star operation * of finite type.


Introduction
The concept of factoring ideals into radical ideals has been studied by various authors. It started with papers by Vaughan and Yeagy [22,23] who studied radical factorization in integral domains. They showed that every integral domain for which every ideal is a finite product of radical ideals (we will call such a domain an SP-domain) is an almost Dedekind domain. Later the first-named author gave a complete characterization in [17] of SP-domains in the context of almost Dedekind domains. After that the second-named author investigated the concept of radical factorization in [19,20] with respect to finitary ideal systems. Further progress in describing SPdomains was made in [8,13,15]. Besides that, radical factorization in commutative rings with identity was investigated in [1]. Many of these results were extended in a recent paper [18] where radical factorization was studied in the context of principally generated C-lattice domains.
Ideal systems of monoids are a generalization of star operations of integral domains. They were studied in detail in [12]. It turns out that (finitary) ideal systems in general fail to be modular (i.e. the lattice of ideals induced by the ideal system is not modular). In particular, an r-SPmonoid (i.e. the "ideal system theoretic analogue" of an SP-domain) can fail to be r-almost Dedekind. The goals of this paper are manyfold. We extend the known characterizations of ralmost Dedekind r-SP-monoids for finitary ideal systems r. We consider lattices of ideals that are (a priori) neither principally generated nor modular. Thus we complement the results of [18] by describing the lattice of r-ideals in case that r is a (not necessarily modular) finitary ideal system. Let p be a modular finitary ideal system and r a finitary ideal system such that every r-ideal is a p-ideal. Then there is a modular finitary ideal system e r p "between" r and p, called the p-modularization of r, which can be used to describe the r-ideals. We show, for instance, that a monoid is an e r p -SP-monoid if and only if every minimal prime s-ideal of a nontrivial r-finitely generated r-ideal is of height one and the radical of every nontrivial principal ideal is r-invertible. We put particular emphasis on the t-system and its modularizations (like the w-system) and present stronger characterizations for these types of ideal systems. As an application we investigate several ring-theoretical constructions with respect to the aforementioned properties.
In Section 2, we introduce the notion of (finitary) ideal systems and most of the important terminology. We also show the most basic properties of the modularizations of a finitary ideal system. In Section 3, we study finitary ideal systems in general. Our main results are characterization theorems for r-almost Dedekind r-SP-monoids as well as r-B ezout r-SP-monoids.
We put our focus on the t-system and its modularizations in Section 4. We will show that a monoid is a w-SP-monoid if and only if the radical of every nontrivial principal ideal is t-invertible. Moreover, we show that a monoid is both a t-B ezout monoid (i.e., a GCD-monoid) and a t-SP-monoid if and only if the radical of every principal ideal is principal.
After that we study the monoid of r-invertible r-ideals in Section 5. In particular, we characterize when every principal ideal of the monoid of r-invertible r-ideals is a finite product of pairwise comparable radical principal ideals. We also give a technical characterization of radical factorial monoids (i.e. monoids for which every principal ideal is a finite product of radical principal ideals). Furthermore, we describe when the monoid of r-invertible r-ideals is radical factorial.
Finally, we investigate several ring-theoretical constructions in Section 6. We show that if R is an integral domain and H is a grading monoid (i.e., a cancellative torsionless monoid), then R½H is a w-SP-domain if and only if R is a w-SP-domain, H is a w-SP-monoid and the homogeneous field of quotients of R½H is radical factorial. We also show that if Ã is a star operation of finite type of an integral domain R, then the Ã -Nagata ring of R is an SP-domain if and only if R is a Ã -almost Dedekind Ã -SP-domain.

Ideal systems
In this section, we introduce the notion of (finitary) ideal systems and the most important terminology. In the following, a monoid H is always a commutative semigroup with identity and more than one element such that every nonzero element of H is cancellative. If not stated otherwise, then H is written multiplicatively.
Throughout this paper let H be a monoid and let G be the quotient monoid of H. Let z(H) denote the set of zero elements of H (i.e., zðHÞ ¼ fz 2 H j zx ¼ z for all x 2 Hg). (We introduce this notion to handle both monoids with and without a zero element. Also note that jzðHÞj 1:) Let X H and Y G: Set ffiffiffi ffi X p ¼ fx 2 H j x n 2 X for some n 2 Ng; called the radical of X and Y À 1 ¼ fz 2 G j zY Hg: We say that X is an s-ideal of H if X ¼ XH [ zðHÞ and we say that X is radical if ffiffiffi ffi X p ¼ X: An s-ideal J of H is called a principal ideal of H if it is generated by at most one element (i.e., J ¼ AH [ zðHÞ for some A H with jAj 1).
If a 2 H; then a is called prime (primary, radical) if aH (i.e. the principal ideal generated by a) is a prime (primary, radical) s-ideal of H. Let XðHÞ denote the set of minimal prime s-ideals of H which properly contain z(H) and let PðXÞ denote the set of prime s-ideals of H that are minimal above X.
By H (resp. H Â ) we denote the set of nonzero elements of H (resp. the set of units of H) and by PðHÞ we denote the power set of H. Let r : PðHÞ ! PðHÞ; X 7 ! X r be a map. For subsets X; Y H and c 2 H we consider the following properties: (A) XH [ zðHÞ X r : (B) If X Y r ; then X r Y r : (C) cX r ¼ ðcXÞ r : (D) X r ¼ [ EX;jEj < 1 E r : We say that r is a (finitary) ideal system on H if r satisfies properties A, B, C (and D) for all X; Y H and c 2 H: Also note that an ideal system r is finitary if and only if X r [ EX;jEj < 1 E r for all X H: Furthermore, if r is an ideal system, then it follows from (A) and (B) that r is idempotent (i.e., ðX r Þ r ¼ X r for each X H).
Let r be finitary ideal system on H and X H: We say that X is an r-ideal (resp. an r-invertible r-ideal) if X r ¼ X (resp. if X r ¼ X and ðXX À 1 Þ r ¼ H). Now let I be an r-ideal of H. The r-ideal I is called nontrivial if zðHÞˆI and it is called proper if IˆH: By I r ðHÞ (resp. I Ã r ðHÞ) we denote the set of r-ideals (resp. the set of r-invertible r-ideals) of H. Observe that ffiffi I p ¼ \ P2PðIÞ P and PðIÞ I r ðHÞ: If I and J are r-ideals of H, then ðIJÞ r is called the r-product of I and J. Note that the set of r-ideals forms a commutative semigroup with identity under r-multiplication and the set of r-invertible r-ideals of H forms a monoid under r-multiplication.
Note that every (nontrivial) principal ideal of H is an (r-invertible) r-ideal of H. Let H be the set of nontrivial principal ideals of H, qðI Ã r ðHÞÞ; resp. qðHÞ; the quotient group of I Ã r ðHÞ; resp. H; and C r ðHÞ ¼ qðI Ã r ðHÞÞ=qðHÞ; called the r-class group of H. Note that C r ðHÞ is trivial if and only if every r-invertible r-ideal of H is principal. Moreover, C r ðHÞ is torsionfree if and only if for all k 2 N and I 2 I Ã r ðHÞ such that ðI k Þ r is principal, it follows that I is principal. Let r-specðHÞ; resp. r-maxðHÞ denote the set of prime r-ideals, resp. the set of r-maximal r-ideals of H. We say that I 2 I r ðHÞ is r-finitely generated if I ¼ E r for some finite E I: We say that r is modular if for all r-ideals I, J, N of H with I N it follows that ðI [ JÞ r \ N ðI [ ðJ \ NÞÞ r (equivalently, for all r-ideals I, J, N of H with I N it follows that ðI [ JÞ r \ N ¼ ðI [ ðJ \ NÞÞ r ). Now let p be a finitary ideal system on H. The ideal system p is called finer than r (or r is called coarser than p), denoted by p r; if X p X r for all X H (equivalently, every r-ideal of H is a p-ideal of H). The notions of finer and coarser can be extended to arbitrary ideal systems.
Next we introduce the most important ideal systems. Let T H be multiplicatively closed (i.e., 1 2 T and xy 2 T for all x; y 2 T). Then there is a unique finitary ideal system T À 1 r defined on T À 1 H such that T À 1 ðX r Þ ¼ ðT À 1 XÞ T À 1 r for all X H: Furthermore, if r is modular, then T À 1 r is modular. If P is a prime s-ideal of H, then we set r P ¼ ðH n PÞ À 1 r: First we define the s-system.
Let s : PðHÞ ! PðHÞ; X 7 ! XH [ zðHÞ: Note that s is a finitary ideal system on H. Next we introduce the v-system and the t-system.
Now let H 6 ¼ G: Let v : PðHÞ ! PðHÞ; X 7 ! ðX À 1 Þ À 1 and t : PðHÞ ! PðHÞ; where R ðXÞ is the ring ideal generated by X. Now let p r: Next we introduce a finitary ideal system e r p depending on p and r. We study some of its elementary properties in Lemma 2.1.
Let e r p : PðHÞ ! PðHÞ; X 7 ! fx 2 H j xF X p and F r ¼ H for some F Hg: Lemma 2.1. Let p and r be finitary ideal systems on H such that p r: (1) e r p is a finitary ideal system on H such that p e r p r: (2) r-maxðHÞ ¼ e r p -maxðHÞ and X e r p ¼ \ M2rÀmaxðHÞ ðX p Þ M for each X H: Then NF Y p and F r ¼ H. w(Claim 1) Claim 2. If X H and x 2 X e r p ; then there are some finite E H and some finite N X such that xE N p ; E r ¼ H, x 2 N e r p and x 2 X r : Let X H and x 2 X e r p : There is some E H such that xE X p and E r ¼ H. Since r is finitary, we can assume without restriction that E is finite. Since p is finitary, there is some finite N X such that xE N p : Consequently, x 2 N e r p and x 2 xH ¼ xE r ¼ ðxEÞ r ðX p Þ r ¼ X r : w(Claim 2) Let X; Y H and c 2 H: If y 2 X p ; then since f1g r ¼ H and y 2 yf1g X p ; we have that y 2 X e r p : Therefore, X p X e r p ; and hence X s X e r p : Next we show that if X Y e r p ; then X e r p Y e r p : Let X Y e r p and x 2 X e r p : By Claim 2 there are some finite E H and some finite N X such that xE N p and E r ¼ H. By Claim 1 there is some F H such that NF Y p and F r ¼ H. This implies that xEF N p F ðN p FÞ p ¼ ðNFÞ p Y p and ðEFÞ r ¼ H; and hence x 2 Y e r p : Now we show that cX e r p ¼ ðcXÞ e r p : First let z 2 X e r p : There is some E H such that zE X p and E r ¼ H. Since czE cX p ¼ ðcXÞ p and E r ¼ H, we have that cz 2 ðcXÞ e r p : Therefore, cX e r p ðcXÞ e r p : Now let z 2 ðcXÞ e r p : It follows by Claim 2 that z 2 ðcXÞ r ¼ cX r ; and hence z ¼ cv for some v 2 H: If c 2 zðHÞ; then z 2 zðHÞ cX e r p : Now let c 6 2 zðHÞ: There is some E H such that cvE ðcXÞ p ¼ cX p and E r ¼ H. Consequently, vE X p ; and thus v 2 X e r p : We infer that z 2 cX e r p : Putting all these parts together shows that e r p is an ideal system on H. We infer by Claim 2 that X e r p [ FX;jFj < 1 F e r p ; and hence e r p is finitary. We have already shown (below the proof of Claim 2) that X p X e r p : Moreover, X e r p X r by Claim 2. This implies that p e r p r: (2) To show that r-maxðHÞ ¼ e r p -maxðHÞ it is sufficient to show that every M 2 e r p -maxðHÞ is an r-ideal of H. Let M 2 e r p -maxðHÞ: Assume that M is not an r-ideal of H. Then M r ¼ H. We have that 1M M p ; and hence 1 2 M e r p ¼ M; a contradiction. Now let X H: Let x 2 X e r p and N 2 r-maxðHÞ: Then xE X p and E r ¼ H for some E H; and thus there is some y 2 E n N: It follows that xy 2 X p ; and hence x 2 y À 1 X p ðX p Þ N : This implies that X e r p ðX p Þ N for every N 2 r-maxðHÞ: Moreover, we have that (3) Since e r p r by (1), we have clearly that I Ã e r p ðHÞ I Ã r ðHÞ: Now let I 2 I Ã r ðHÞ: Assume that I 6 2 I Ã e r p ðHÞ: Then ðII À 1 Þ e r pˆH ; and hence there is some M 2 e r p -maxðHÞ such that II À 1 M: We infer by (2) that M 2 r-maxðHÞ; and hence H ¼ ðII À 1 Þ r M r ¼ M; a contradiction. It remains to show that the e r p -multiplication and the r-multiplication coincide on I Ã e r p ðHÞ: Let J; L 2 I Ã e r p ðHÞ: Then ðJLÞ e r p 2 I Ã e r p ðHÞ I r ðHÞ: We infer that ðJLÞ e r p ¼ ððJLÞ e r p Þ r ¼ ðJLÞ r ; since e r p r by (1 (5) Let m and n be finitary ideal systems on H such that p m e r p n r and X H: First let x 2 X e n m : Then there is some finite E H such that xE X m and E n ¼ H. Then xE X e r p and E r ¼ H. As shown in (1), we have that xEF X p for some F H with F r ¼ H.
Observe that ðEFÞ r ¼ H; and thus x 2 X e r p : Now let x 2 X e r p : Then xE X p and E r ¼ H for some E H: By (2) we have that E e r p ¼ H: Therefore, xE X m and E n ¼ H, and hence x 2 X e n m : w If p r are finitary ideal systems on H and p is modular, then we say (in view of Lemma 2.1(4)) that e r p is the p-modularization of r. Set w p ¼ e t p and w ¼ w s . We have that s is the finest ideal system on H, t is the coarsest finitary ideal system on H and v is the coarsest ideal system on H. Furthermore, s w t v and if H is an integral domain, then s d w d t v: Note that both the s-system and the d-system are modular finitary ideal systems. In what follows, we use the remarks of this paragraph without further citation.

Results for finitary ideal systems
Let r be a finitary ideal system on H. We say that H is an r-SP-monoid if every r-ideal of H is a finite r-product of radical r-ideals of H. Moreover, H is called radical factorial if every principal ideal of H is a finite product of radical principal ideals of H. Furthermore, H is called factorial if every principal ideal of H is a finite product of prime principal ideals of H (equivalently, every nontrivial prime t-ideal of H contains a nontrivial prime principal ideal of H). We say that H is a valuation monoid if the principal ideals of H are pairwise comparable (equivalently, the s-ideals of H are pairwise comparable). Also note that if H is a valuation monoid, then s ¼ r ¼ t (i.e., the s-system is the unique finitary ideal system on H). Moreover, if H 6 ¼ G; then H is called a discrete valuation monoid (or a DVM) if every s-ideal of H is principal (equivalently, every prime s-ideal of H is principal). We say that H satisfies the Principal Ideal Theorem if for each nontrivial principal ideal I of H we have that PðIÞ XðHÞ: Finally, H is called r-local if H n H Â is an r-ideal of H (equivalently, jr-maxðHÞj ¼ 1Þ: Observe that if H is r-local, then C r ðHÞ is trivial. It is easy to see that if the radical of every nontrivial principal ideal of H is r-invertible or every nontrivial principal ideal of H is a finite r-product of radical r-ideals of H (in particular if H is radical factorial or an r-SP-monoid), then H M is radical factorial for each M 2 r-maxðHÞ: (In the first case we can show that the radical of every principal ideal of H M is principal for each M 2 r-maxðHÞ and then apply [19,Proposition 2.10].) The main purpose of this section is to present new characterizations of r-almost Dedekind monoids, r-almost Dedekind r-SP-monoids and r-B ezout r-SP-monoids.
Finally we show that PðIÞ XðHÞ for each r-invertible r-ideal I of H. Let I be an r-invertible r-ideal of H and P 2 PðIÞ: There is some M 2 r-maxðHÞ such that P M: Observe that I M is a nontrivial principal ideal of H M and P M 2 PðI M Þ XðH M Þ: Therefore, there is some P 0 2 XðHÞ such that P 0 M and P M ¼ ðP 0 Þ M : This implies that P ¼ P M \ H ¼ ðP 0 Þ M \ H ¼ P 0 2 XðHÞ: w Let r be a finitary ideal system on H. The monoid H is called r-treed if for all M 2 r-maxðHÞ; it follows that the prime r-ideals of H that are contained in M form a chain. Moreover, H is called an r-almost Dedekind monoid (or an almost r-Dedekind monoid in the terminology of [19]) if H ¼ G or if H M is a DVM for each M 2 r-maxðHÞ: Lemma 3.2. Let r be a finitary ideal system on H such that every nontrivial prime r-ideal of H contains an r-invertible radical r-ideal of H.
(1) If the prime r-ideals of H form a chain and H 6 ¼ G, then H is a DVM.
(2) If H is r-treed, then H is an r-almost Dedekind r-SP-monoid.

Proof.
(1) Let the prime r-ideals of H form a chain and let H 6 ¼ G: Then H is r-local, and thus every rinvertible r-ideal of H is principal. Moreover, every radical r-ideal of H is a prime r-ideal of H. Therefore, every nontrivial prime r-ideal of H contains a nontrivial prime principal ideal of H. Let X be the set of all elements of H which can be represented as a product of a unit of H times a (possibly empty) finite product of nonzero prime elements of H. Assume that H is not factorial. Then there is some nonzero x 2 H n X: It is straightforward to show that xH \ X ¼ ;: Since X is a multiplicatively closed subset of H, xH is an r-ideal of H and r is finitary, we infer that xH P and P \ X ¼ ; for some prime r-ideal P of H. Since P contains a nonzero prime principal ideal of H, we have that P \ X 6 ¼ ;; a contradiction. This implies that H is a factorial monoid. Since the prime r-ideals of H form a chain, we have that jXðHÞj ¼ 1; and thus H is a DVM. Proof. Suppose to the contrary that there is some nonzero radical element x 2 H such that x k 2 ð Q kþ1 i¼1 I i Þ r : We infer that x 2 P: It follows that P P ¼ xH P ¼ ðI j Þ P for each j 2 ½1; k þ 1; and hence P k w Proposition 3.4. Let r be a finitary ideal system on H and let the radical of every principal ideal of H be principal.
(1) For each nontrivial r-finitely generated r-ideal I of H there is some nonzero z 2 H such that fP 2 XðHÞ j I Pg ¼ fP 2 XðHÞ j z 2 Pg: (2) C r ðHÞ is trivial.

Proof.
(1) Claim 1. If a, b are nonzero radical elements of H such that b divides a, then To prove Claim 1 let a; b 2 H be nonzero radical elements of H such that b divides a. First let P 2 XðHÞ be such that a b 2 P: It is obvious that a 2 P: Since aH P is a nonzero radical ideal of H P we have that aH P ¼ P P . Suppose that b 2 P: Then a 2 P 2 ; and hence aH P ¼ P P ¼ P 2 P ¼ a 2 H P : Therefore, a 2 H Â P ; a contradiction. We infer that b 6 2 P: The converse inclusion is trivially satisfied.
w(Claim 1) Claim 2. For all nonzero x; y 2 H there is some nonzero z 2 H such that fP 2 XðHÞ j ðxH [ yHÞ r Pg ¼ fP 2 XðHÞ j z 2 Pg: To prove Claim 2 let x; y 2 H be nonzero. There exist nonzero radical elements a; b; c 2 H such that ffiffiffiffiffiffiffiffi ffi XðHÞ j a c 2 Pg ¼ fP 2 XðHÞ j a 2 P; ðb 6 2 P or c 6 2 PÞg: We infer by Claim 1 that fP 2 XðHÞ j x; y 2 Pg ¼ fP 2 XðHÞ j b; c 2 Pg ¼ fP 2 XðHÞ j a 2 P; d 6 2 Pg ¼ fP 2 XðHÞ j z 2 Pg: w(Claim 2) The statement now follows by induction from Claim 2.
(2) Claim. The radical of every r-invertible r-ideal of H is principal.
To prove the claim let I be an r-invertible r-ideal of H. By Proposition 3.1, we have that PðIÞ XðHÞ: It follows by (1) that there is some nonzero z 2 H such that PðIÞ ¼ fP 2 XðHÞ j I Pg ¼ fP 2 XðHÞ j z 2 Pg ¼ PðzHÞ; and hence ffiffi Now let J be an r-invertible r-ideal of H. By the claim there is some nonzero radical z 1 2 H such that ffiffi J p ¼ z 1 H: Therefore, z k 1 2 J for some k 2 N: Next we recursively construct nonzero radical elements and suppose that we have already constructed the first i elements. It follows that This completes the construction. Assume that z kþ1 6 2 H Â : Then there is some P 2 Pðz kþ1 HÞ: It follows from Proposition 3.1 that P 2 XðHÞ: Observe that z i H z iþ1 H for each i 2 ½1; k; and thus [ kþ1 i¼1 z i H P: Moreover, we have that z k j¼1 z j HÞ r ; which contradicts Lemma 3.3. Therefore, z kþ1 2 H Â ; and hence w Proposition 3.5. Let H 6 ¼ G and r a finitary ideal system on H and let H be r-local such that the radical of every r-finitely generated r-ideal of H is principal. Then H is a DVM.
Proof. By Proposition 3.1, H satisfies the Principal Ideal Theorem. Assume that H is not a valuation monoid. Then there exist x; y 2 H such that xH6 yH and yH6 xH: Using the fact that the radical of every r-finitely generated r-ideal of H is principal, we can recursively construct nonzero radical elements z i of H such that for every i 2 N; Let r be a finitary ideal system on H. We say that H satisfies the r-prime power condition if every primary r-ideal of H is an r-power of its radical. Note that every r-SP-monoid satisfies the r-prime power condition (see [19,Proposition 3.10 (1)]). Moreover, H satisfies the strong r-prime power condition if every r-ideal of H with prime radical is an r-power of its radical. Finally, H is called primary r-ideal inclusive if for all P; Q 2 r-specðHÞ such that PˆQ it follows that P Iˆffi ffi I p Q for some primary r-ideal I of H. Now let I be an r-ideal of H. We say that I is r-cancellative if for all r-ideals J and L of H such that ðIJÞ r ¼ ðILÞ r it follows that J ¼ L. Moreover, I is called r-half cancellative (or r-unit-cancellative) if for all J 2 I r ðHÞ with I ¼ ðIJÞ r it follows that J ¼ H.
Let T H a multiplicatively closed subset. Note that if H satisfies the (strong) r-prime power condition, then T À 1 H satisfies the (strong) T À 1 r-prime power condition. Moreover, if H is primary r-ideal inclusive, then T À 1 H is primary T À 1 r-ideal inclusive. (By [19,Lemma 3.8] it remains to show that if H satisfies the strong r-prime power condition, then T À 1 H satisfies the strong T À 1 r-prime power condition. Let H satisfy the strong r-prime power condition and let J be a T À 1 r-ideal of T À 1 H with prime radical. Set I ¼ J \ H: Then I is an r-ideal of H and J ¼ In what follows, we use the remarks of this paragraph without further citation. (1) H is an r-almost Dedekind monoid.
(2) H satisfies the strong r-prime power condition and every nontrivial r-ideal of H is rcancellative.
(3) For all nonzero x 2 H and P 2 PðxHÞ, P is r-half cancellative and every r-ideal of H whose radical is P is an r-power of its radical. (4) H is r-treed and satisfies the strong r-prime power condition.
(5) H satisfies the strong r-prime power condition and r is modular. (6) r-maxðHÞ ¼ XðHÞ and H satisfies the r-prime power condition. (7) H satisfies the r-prime power condition and the Principal Ideal Theorem, and H is primary rideal inclusive.
Proof. Claim 1. If H satisfies the r-prime power condition and P 2 PðxHÞ for some nonzero x 2 H; then P P is principal. Let x 2 H be nonzero and P 2 PðxHÞ: Since P P is the only prime s-ideal of H P such that x 2 P P ; we infer that ffiffiffiffiffiffiffiffi xH P p ¼ P P ; Note that H P satisfies the r P -prime power condition (by the discussion above), and thus xH P ¼ ðP k P Þ r P for some k 2 N: (Note that P P 2 r P -maxðH P Þ; and thus xH P is P P -primary.) Therefore, P P is r P -invertible, and hence P P is principal, since H P is r P -local.
w(Claim 1) (1) ) (2),(5): Clearly, r-maxðHÞ ¼ XðHÞ: Let I be a nontrivial r-ideal of H and J, L r-ideals of H such that ðIJÞ r ¼ ðILÞ r : If M 2 r-maxðHÞ; then I M ¼ xH M for some nonzero x 2 H M and hence Observe that H M is a DVM, and thus every nontrivial s-ideal of H M is a power of M M . Consequently, (2) ) (3): This is obvious.
(3) ) (4): It is sufficient to show that every r-maximal r-ideal of H is of height one. Let M be an r-maximal r-ideal of H. Assume that M is not of height one, then there exist x 2 M n f0g and P 2 PðxHÞ such that PˆM: By Claim 1 there is some y 2 P P such that P P ¼ yH P . We have that ffiffiffiffiffiffiffiffiffiffiffiffi ffi and thus ðPMÞ r ¼ ðP k Þ r for some k 2 N: Since ðP 2 Þ r ðPMÞ r P and P is r-half cancellative, we infer that ðPMÞ r ¼ ðP 2 Þ r ; and thus yH P ¼ ðP P M P Þ r P ¼ ðP 2 P Þ r P ¼ y 2 H P : This implies that P P ¼ yH P ¼ H P ; a contradiction. (5) ) (6): Assume that r-maxðHÞ 6 ¼ XðHÞ: Then there exist y 2 H ; P 2 PðyHÞ and M 2 r-maxðHÞ such that PˆM: By Claim 1 there is some x 2 P such that P P ¼ xH P . Observe that ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ððP 2 Þ r [ xMÞ r p ¼ P; and hence ððP 2 Þ r [ xMÞ r ¼ ðP k Þ r for some k 2 N: If k ! 2; then xM ðP 2 Þ r ; and thus xH P ¼ xM P ðP 2 P Þ r P ¼ x 2 H P ; a contradiction. Therefore, ððP 2 Þ r [ xMÞ r ¼ P: If z 2 H is such that xz 2 ðP 2 Þ r ; then xz 2 ððP 2 Þ r Þ P ¼ x 2 H P ; and thus z 2 xH P \ H ¼ P: We infer that (6) ) (7): It is clear that H satisfies the Principal Ideal Theorem. It follows from [19,Proposition 3.9] that H is primary r-ideal inclusive.
(7) ) (1): Recall that a prime r-ideal P of H is called r-branched if there exists a P-primary rideal I of H with I 6 ¼ P: Claim 2. For each r-branched prime r-ideal P of H, we have that P 2 XðHÞ and H P is a DVM. Let P be an r-branched prime r-ideal of H. Then P P is a principal ideal of H P by [19,Proposition 5.2(1)]. There is some x 2 H such that P P ¼ xH P . Observe that P 2 PðxHÞ XðHÞ; and hence P P 2 XðH P Þ: Therefore, every prime s-ideal of H P is principal, and hence H P is a DVM.
w(Claim 2) Let M 2 r-maxðHÞ: It is sufficient to show that M 2 XðHÞ (then M is r-branched, and hence H M is a DVM by Claim 2). Assume that M 6 2 XðHÞ: Then there is some nontrivial prime r-ideal P of H such that PˆM: Since H is primary r-ideal inclusive, we can find an r-branched prime r-ideal Q of H such that PˆQ: By Claim 2 we have that Q 2 XðHÞ; a contradiction. w Lemma 3.7. Let r be a modular finitary ideal system on H. Then H is primary r-ideal inclusive.
Proof. Let P; Q 2 r-specðHÞ be such that PˆQ: There exist x 2 Q n P and L 2 PððP [ x 2 HÞ r Þ such that L Q: Set I ¼ ððP [ x 2 HÞ r Þ L \ H: Observe that I is an L-primary r-ideal of H. It remains to show that I 6 ¼ L: Assume to the contrary that  Now let the equivalent conditions be satisfied. Since e r p r t; it is sufficient to show that every e r p -ideal of H is a t-ideal of H. Let I 2 I e r p ðHÞ: Observe that H is an e r p -Pr€ ufer monoid, and hence every e r p -finitely generated e r p -ideal of H is a t-ideal of H. Since e r p is finitary, we infer that I is a directed union of t-ideals of H, and hence I is a t-ideal of H. w Theorem 3.9. Let r be a finitary ideal system on H. The following are equivalent: (1) H is an r-almost Dedekind r-SP-monoid.
(3) ) (1): First we show that H satisfies the Principal Ideal Theorem. Let x 2 H and P 2 PðxHÞ: It follows by Claim 1 in the proof of Proposition 3.6 that P P is principal. Observe that every nontrivial prime r P -ideal of H P contains a nontrivial radical principal ideal of H P , and thus P P 2 XðH P Þ by [19, Lemma 2.3(2)] (since P P is principal and thus minimal above a nontrivial radical principal ideal of H P ). Therefore, P 2 XðHÞ: Consequently, H is an r-almost Dedekind monoid by Proposition 3.6. By [19,Corollary 3.4] we have that H is an r-SP-monoid.
is principal. It follows by Proposition 3.5 that H M is a DVM. We infer that H is an r-almost Dedekind monoid. It follows by [19,Corollary 3.4] that H is an r-SP-monoid. (1) H is an r-B ezout r-SP-monoid.
(3) H is r-treed, C r ðHÞ is trivial and every nontrivial prime r-ideal of H contains a nontrivial radical principal ideal of H. (4) H satisfies the r-prime power condition, H is primary r-ideal inclusive and the radical of every principal ideal of H is principal. (5) H is r-treed and the radical of every principal ideal of H is principal. (6) PðIÞ XðHÞ for every nontrivial r-finitely generated r-ideal I of H and the radical of every principal ideal of H is principal.
The radical of every r-finitely generated r-ideal of H is principal.
(2) ) (3): Since H is an r-B ezout monoid, it is clear that H is r-treed and C r ðHÞ is trivial. Since H is radical factorial, every nontrivial prime r-ideal of H contains a nontrivial radical principal ideal of H.
(3) ) (1): This is an immediate consequence of Theorem 3.9, since every r-almost Dedekind monoid with trivial r-class group is an r-B ezout monoid.
(1) ) (4): It follows from Theorem 3.9 that H satisfies the r-prime power condition, that H is primary r-ideal inclusive and that the radical of every nontrivial principal ideal of H is r-invertible. Since H is an r-B ezout monoid, we infer that the radical of every principal ideal of H is principal.
(6) ) (7): Let I be a nontrivial r-finitely generated r-ideal of H. By Proposition 3.4(1), we have that PðIÞ ¼ fP 2 XðHÞ j I Pg ¼ fP 2 XðHÞ j z 2 Pg ¼ PðzHÞ for some nonzero z 2 H: Consequently, ffiffi I p ¼ ffiffiffiffiffiffi zH p is principal. (7) ) (1): By Theorem 3.9, H is an r-almost Dedekind r-SP-monoid. We infer by Proposition 3.4(2) that H is an r-B ezout monoid. w Next we rediscover several well-known characterizations for (B ezout) SP-domains and we also present some new characterizations. (1) R is an SP-domain.
(2) R is treed and every nonzero prime ideal of R contains an invertible radical ideal of R.
(3) Every primary ideal of R is a power of its radical and every nonzero prime ideal of R contains an invertible radical ideal of R. (4) Every minimal prime ideal of each nonzero finitely generated ideal of R is of height one and the radical of every nonzero principal ideal of R is invertible. (1) R is a B ezout SP-domain.
(2) R is a radical factorial B ezout domain.
(3) R is treed and the radical of every principal ideal of R is principal.
(4) Every primary ideal of R is a power of its radical and the radical of every principal ideal of R is principal. (5) Every minimal prime ideal of each nonzero finitely generated ideal of R is of height one and the radical of every principal ideal of R is principal. (6) The radical of every finitely generated ideal of R is principal.
Proof. This is an easy consequence of Lemma 3.7 and Theorems 3.9 and 3.10.
w Note that there are examples of t-SP-monoids that fail to be t-almost Dedekind monoids. As shown in [20,Example 4.2] there is some t-local t-SP-monoid H such that every nontrivial t-ideal of H is t-cancellative and t-dimðHÞ ¼ 2: In particular, H satisfies the t-prime power condition and PðIÞ XðHÞ for each nontrivial t-finitely generated t-ideal of H. Note that H does not satisfy the strong t-prime power condition, H is not t-treed and H is not primary t-ideal inclusive.

On the t-system and the w-system
In this section, we study the t-system and its modularizations. We present stronger characterizations for these types of finitary ideal systems than in the section before. Besides that, we investigate the connections with the modularizations e r p of a finitary ideal system r in general and describe e r p -SP-monoids and e r p -B ezout e r p -SP-monoids. We also show that the t-class group of every radical factorial BF-monoid is torsionfree. Let r be a finitary ideal system on H. We say that H is an r-finite conductor monoid if xH \ yH is r-finitely generated for all x; y 2 H: Proposition 4.1. [cf. [11,24]] Let P be a set of prime s-ideals of H such that \ P2P H P ¼ H and H Q is a valuation monoid for every Q 2 P. Let I and J be t-ideals of H.
(1) If I, J and I \ J are t-finitely generated, then ðIJÞ t ¼ ððI \ JÞðI [ JÞ t Þ t : (2) If I and J are t-invertible and I \ J is t-finitely generated, then I \ J and ðI [ JÞ t are t-invertible. (3) If H is a t-finite conductor monoid, then H is a t-Pr€ ufer monoid.
Proof. Observe that r : PðHÞ ! PðHÞ defined by X r ¼ \ P2P ðX s Þ P for each X H is an ideal system on H. This implies that r v; and hence I ¼ \ P2P I P for each divisorial ideal I of H.
(1) Let I, J and I \ J be t-finitely generated. Then ðIJÞ t and ðI [ JÞ t are t-finitely generated.
This implies that ððI \ JÞðI [ JÞÞ t ¼ ððI \ JÞðI [ JÞ t Þ t is t-finitely generated. Therefore, it is sufficient to show that ðððI \ JÞðI [ JÞÞ t Þ P ¼ ððIJÞ t Þ P for each P 2 P: Let P 2 P: Since H P is a valuation monoid, we have that I P J P or J P I P : Consequently, ðððI \ JÞðI [ JÞÞ t Þ P ¼ ððI P \ J P ÞðI P [ J P ÞÞ t P ¼ ðI P J P Þ t P ¼ ððIJÞ t Þ P : (2) Let I and J be t-invertible and let I \ J be t-finitely generated. Clearly, I and J are t-finitely generated, and thus ððI \ JÞðI [ JÞ t Þ t ¼ ðIJÞ t : Since ðIJÞ t is t-invertible, we have that ððI \ JÞðI [ JÞ t Þ t is t-invertible, and hence I \ J and ðI [ JÞ t are t-invertible.

(3) Let H be a t-finite conductor monoid. First we show that for each nonempty finite A H
and each x 2 H it follows that A t \ xH is t-finitely generated. Let A H be finite and nonempty and x 2 H: Let P 2 P: Since H P is a valuation monoid, we have that ðA t Þ P ¼ AH P : Next we show by induction that for each n 2 N and all E H with jEj ¼ n it follows that E t is t-invertible. The statement is clearly true for n ¼ 1. Now let n 2 N and F H be such that jFj ¼ n þ 1: There exist E F and x 2 F n E such that F ¼ E [ fxg and jEj ¼ n: It follows by the previous claim that E t \ xH is t-finitely generated. We infer by (2) that The following are equivalent: (1) H is a t-almost Dedekind t-SP-monoid.
(2) H is a t-finite conductor monoid and every principal ideal of H is a finite t-product of radical t-ideals of H. (3) Every t-ideal of H is a t-product of finitely many pairwise comparable radical t-ideals of H. (4) The radical of every nontrivial principal ideal of H is t-invertible.
(2) ) (1): By Proposition 3.1 we have that \ P2XðHÞ H P ¼ H and H Q is a DVM for every Q 2 XðHÞ: It follows by Proposition 4.1(3) that H is a t-Pr€ ufer monoid, and hence H is t-treed. Consequently, H is a t-almost Dedekind t-SP-monoid by Theorem 3.9.
(3) ) (4): Let x 2 H : There exist n 2 N and finitely many radical t-ideals I i of H such that I i I iþ1 for each i 2 ½1; n À 1 and xH ¼ ð Q n i¼1 I i Þ t : This implies that (4) ) (1): By Theorem 3.9 it is sufficient to show that the radical of every nontrivial t-finitely generated t-ideal of H is t-invertible.
It follows by Proposition 3.1 that \ P2XðHÞ H P ¼ H; H Q is a DVM for each Q 2 XðHÞ and PðAÞ XðHÞ for each t-invertible t-ideal A of H.
It is sufficient to show by induction that for each n 2 N and each E H with jEj ¼ n it follows that ffiffiffiffi ffi E t p is t-invertible. The statement is clearly true for n ¼ 1. Now let n 2 N and F H be such that jFj ¼ n þ 1: There exist E F and x 2 F n E such that jEj ¼ n and F ¼ E where the first equality holds since PððI [ JÞ t Þ XðHÞ; and the last equality holds since ðI [ JÞ t is t-finitely generated (and hence divisorial). Therefore, (1) H is an r-almost Dedekind r-SP-monoid.
(2) r-maxðHÞ ¼ t-maxðHÞ and the radical of every nontrivial principal ideal of H is t-invertible. (1) H is an r-B ezout r-SP-monoid.
(2) r-maxðHÞ ¼ t-maxðHÞ and the radical of every principal ideal of H is principal.
(3) H is an e r p -B ezout e r p -SP-monoid.
Proof. (A) (1) ) (2): First let H be an r-almost Dedekind r-SP-monoid. Clearly, r-maxðHÞ ¼ XðHÞ; and since every height-one prime s-ideal of H is a t-ideal, we infer that r-maxðHÞ ¼ t-maxðHÞ: By Theorem 3.9, the radical of every nontrivial principal ideal of H is r-invertible. Since r t; we have that the radical of every nontrivial principal ideal of H is t-invertible.
(2) ) (1): Now let r-maxðHÞ ¼ t-maxðHÞ and let the radical of every nontrivial principal ideal of H be t-invertible. It follows by Theorem 4.2 that H is a t-almost Dedekind monoid, and hence r-maxðHÞ ¼ t-maxðHÞ ¼ XðHÞ: Therefore, H is r-treed and every t-invertible t-ideal of H is an r-invertible r-ideal of H. Consequently, H is an r-almost Dedekind r-SP-monoid by Theorem 3.9.
(2) () (3): By Lemmas 2.1(4) and 3.7, Theorem 3.9 and [19, Proposition 3.10(1)], we have that H is an e r p -SP-monoid if and only if H is an e r p -almost Dedekind e r p -SP-monoid. Now applying the equivalence of (1) and (2) to e r p and using the fact that r-maxðHÞ ¼ e r p -maxðHÞ gives us the desired equivalence.
w Corollary 4.4. The following are equivalent: (1) H is a t-almost Dedekind t-SP-monoid.
(3) H is a w-finite conductor monoid and every principal ideal of H is a finite w-product of radical w-ideals of H. (4) Every w-ideal of H is a w-product of finitely many pairwise comparable radical w-ideals of H. (5) The radical of every nontrivial principal ideal of H is w-invertible.
Proof. (1) ) (2): By Theorem 4.2, the radical of every nontrivial principal ideal is t-invertible. As pointed out before, we have that w-maxðHÞ ¼ t-maxðHÞ: We infer by Theorem 4.3(A) that H is a w-almost Dedekind w-SP-monoid.
(3) ) (1): Let x; y 2 H: Then xH \ yH ¼ E w for some finite E H: Since w t; we infer that xH \ yH ¼ ðxH \ yHÞ t ¼ ðE w Þ t ¼ E t : Therefore, H is a t-finite conductor monoid. Note that every nontrivial principal ideal of H is a finite w-product of w-invertible radical w-ideals of H. Therefore, every nontrivial principal ideal of H is a finite t-product of (t-invertible) radical tideals of H by Lemma 2.1(3). The statement now follows from Theorem 4.2.
(2) H is a w-B ezout w-SP-monoid.  (2)] that H satisfies the Principal Ideal Theorem. Let x 2 H be nonzero. Clearly, there is a sequence ðz i Þ i2N of nonzero radical elements of H such that ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ðx= Q ' À 1 i¼1 z i ÞH q ¼ z ' H for each ' 2 N: Moreover, we have that z ' H z 'þ1 H for all ' 2 N: Since ffiffiffiffiffiffi ffi xH p ¼ z 1 H; there is some k 2 N such that z k 1 2 xH: We infer by Lemma 3.3 that z kþ1 2 H Â ; and thus xH ¼ Q k i¼1 z i H: (4) ) (3): Let x 2 H : Then there exist n 2 N and finitely many radical principal ideals I i of H such that xH ¼ Q n i¼1 I i and I i I iþ1 for all i 2 ½1; n À 1: It follows that  Proof. This is an immediate consequence of Theorem 3.10, Corollary 4.5 and [19, Theorem 2.14]. w Finally, we give a partial positive answer to the (so far) unsolved problem of whether the tclass group of a radical factorial monoid is torsionfree. The following result shows that the t-class group of a radical factorial monoid has to satisfy a "weak form" of being torsionfree. Let H be a monoid and A PðHÞ: A function k : A ! N 0 is called a length function on A if kðJÞ < kðIÞ for all I; J 2 A with IˆJ: Moreover, H is called a BF-monoid if the set of nontrivial principal ideals of H possesses a length function.
Proposition 4.7. Let H be a radical factorial monoid, k 2 N; I 2 I Ã t ðHÞ such that ðI k Þ t is principal and A ¼ fðL k Þ t j L 2 I Ã t ðHÞ; I L; ðL k Þ t is principalg: (1) If A possesses a length function, then I is principal.
(2) If fP 2 XðHÞ j I Pg is finite, then I is principal.
(3) If H is a BF-monoid, then C t ðHÞ is torsionfree. Proof.
(1) Let k : A ! N 0 be a length function on A: It is sufficient to show by induction that for each n 2 N 0 and L 2 I Ã t ðHÞ such that I L; ðL k Þ t is principal and kððL k Þ t Þ ¼ n; it follows that L is principal. Let n 2 N 0 and L 2 I Ã t ðHÞ be such that I L; ðL k Þ t is principal and kððL k Þ t Þ ¼ n: Without restriction let L 6 ¼ H: There is some radical nonunit x 2 H such that xH: Consequently, L ¼ xJ for some J 2 I Ã t ðHÞ: Note that ðJ k Þ t is principal, I J and ðL k Þ tˆð J k Þ t : We infer that kððJ k Þ t Þ < n; and hence J is principal by the induction hypothesis. This implies that L is principal.
(2) Let fP 2 XðHÞ j I Pg be finite. Let P be the set of all finite t-products (which are not necessarily squarefree or nonempty) of elements of XðHÞ: Since H Q is a DVM for each Q 2 XðHÞ by Proposition 3.1, we infer that fC 2 P j ðI k Þ t Cg is finite. Let k : A ! N 0 be defined by kðLÞ ¼ jfC 2 P j L Cgj for each L 2 A: Now let A; B 2 A be such that AˆB: There exist x; y 2 H and some nonunit z 2 H such that A ¼ xH, B ¼ yH and x ¼ yz. Since H satisfies the Principal Ideal Theorem by Proposition 3.1, there is some Q 2 XðHÞ such that z 2 Q: Moreover, there is some minimal J 2 P such that yH J: We have that A ¼ xH ðJQÞ t 2 P and B ¼ yH6 ðJQÞ tˆJ (note that H Q is a DVM). Therefore, kðBÞ < kðAÞ; and thus k is a length function. The statement now follows by (1). (3) Let H be a BF-monoid, ' 2 N and L 2 I Ã t ðHÞ such that ðL ' Þ t is principal. Set B ¼ fðJ ' Þ t j J 2 I Ã t ðHÞ; L J; ðJ ' Þ t is principalg: Since B is a subset of the set of nontrivial principal ideals of H, we have that B possesses a length function. Therefore, L is principal by (1). We infer that C t ðHÞ is torsionfree.

On the monoid of r-invertible r-ideals
In this section, we put our focus on the monoid of r-invertible r-ideals and give characterizations for this monoid to be radical factorial or to have the property that the radical of every principal ideal is principal. We also present a characterization for radical factorial monoids and discuss the connections between the monoid of r-invertible r-ideals and radical r-factorization of principal ideals and r-invertible r-ideals. (1) Let J be an r-invertible r-ideal of H. If I divides J in I Ã r ðHÞ; then J ¼ ðIAÞ r for some r-invertible r-ideal A of H, and thus J ¼ ðIAÞ r ðIHÞ r ¼ I: Conversely, if J I; then B ¼ ðJI À 1 Þ r is an r-invertible r-ideal of H and J ¼ ðBIÞ r ; and hence I divides J in I Ã r ðHÞ: (2) First let I be radical, J 2 I Ã I ðP m Þ r : Then ð Q J2X J k Þ r ðP m Þ r ; and hence P k' P ¼ ðð Q J2X J k Þ r Þ P ððP m Þ r Þ P ¼ P m P : Since k' < m this contradicts the fact that P P is a nontrivial proper principal ideal of H P .

w(Claim)
For A 2 I Ã r ðHÞ; we set m A ¼ maxfk 2 N 0 j A ðP k Þ r for some P 2 XðHÞg (which exists by the claim). It is sufficient to show by induction that for all m 2 N 0 and I 2 I Ã r ðHÞ with m I ¼ m, that I is a finite r-product of radical r-ideals of H.
Let m 2 N 0 and I 2 I Ã r ðHÞ be such that m I ¼ m. If m ¼ 0, then since I is divisorial, I ¼ \ P2XðHÞ I P ¼ \ P2XðHÞ H P ¼ H and we are done. Now let m > 0. There is some (r, I)-meager set X I Ã r ðHÞ such that ffiffi I p ¼ \ J2X J: As in the proof of the claim, it follows that each element of X is a radical r-ideal of H.
Let P 2 XðHÞ and set ' ¼ jfJ 2 X j J Pgj: Then ðð Q J2X JÞ r Þ P ¼ Q J2X J P ¼ P ' P ¼ ððP ' Þ r Þ P I P : Since I and ð Q J2X JÞ r are divisorial, we have that I ð Q J2X JÞ r : We infer that I ¼ ðL Q J2X JÞ r for some r-invertible r-ideal L of H. It is sufficient to show that m L < m: Without restriction let m L > 0: There is some Q 2 XðHÞ such that m L ¼ maxfk 2 N 0 j L ðQ k Þ r g: Since m L > 0; we have that I L Q; and thus J Q for some J 2 X: Since I ðJLÞ r ðQ m L þ1 Þ r ; we infer that m L < m L þ 1 m: (B) This can be shown along the same lines as "(A) (2) () (A) (3)", by replacing r with t and by replacing r-invertible r-ideals with nontrivial principal ideals. Proof. Note that every r-almost Dedekind monoid is an r-Pr€ ufer monoid and every r-Pr€ ufer monoid is r-treed. Moreover, if every r-invertible r-ideal of H is a finite r-product of radical r-ideals of H, then clearly every nontrivial prime r-ideal of H contains an r-invertible radical r-ideal of H. Therefore, the equivalence is an immediate consequence of Theorem 3.9 and Proposition 5.2(A). w finitely generated, there is some n 2 N such that ðJ n Þ r I: Therefore, I divides ðJ n Þ r in I by Lemma 5.1(1), and hence ðJIÞ n ¼ ðJ n Þ r I II : This implies that JI ffiffiffiffiffi II p : (4) ) (5): Let I be an r-invertible r-ideal of H. Set I ¼ I Ã r ðHÞ: By Corollary 4.5, there exist n 2 N and finitely many radical elements I i of I such that II ¼ Q n i¼1 I i I ¼ ð Q n i¼1 I i Þ r I and I i I I iþ1 I for all i 2 ½1; n À 1: This implies that I ¼ ð Q n i¼1 I i Þ r : Let i 2 ½1; n: It follows by Lemma 5.1(2) that I i is a radical r-ideal of H. Furthermore, if i 2 ½1; n À 1; then I i I iþ1 by Lemma 5.1(1).
(5) ) (1): This is obvious. w 6. Monoid rings and Ã-Nagata rings In this section, let H always be a monoid with zðHÞ ¼ ;: As an application, we study several ring-theoretical constructions in this section. Recall that the monoid H is completely integrally closed if for all x 2 H and y 2 G with xy n 2 H for all n 2 N; it follows that y 2 H: Moreover, H is called root-closed if for all x 2 G and n 2 N with x n 2 H; we have that x 2 H: We say that H is a grading monoid if H is torsionless (i.e., for all x; y 2 H and n 2 N such that x n ¼ y n it follows that x ¼ y). If not stated otherwise, we will write a grading monoid additively (from now on). Note that H is a grading monoid if and only if we can define a total order on it which is compatible to the monoid operation ([16, page 123]). Moreover, a nontrivial Abelian group is a grading monoid if and only if it is torsionfree. Let R be an integral domain, H a grading monoid, K be a field of quotients of R and G a quotient group of H. A sequence ðx g Þ g2I of elements of K is called formally infinite if all but finitely many elements of that sequence are zero.
By R½H ¼ R½X; H ¼ f P g2H x g X g j ðx g Þ g2H 2 R H is formally infiniteg we denote the monoid ring over R and H. It is well-known that R½H is an integral domain. Note that R½H is integrally closed if and only if R is integrally closed and H is root-closed [2, Theorem 3.7(d)]. Furthermore, R½H is completely integrally closed if and only if R and H are completely integrally closed [2, Theorem 3.7(e)]. If B K and Y G; then set B½Y ¼ f P g2Y x g X g j ðx g Þ g2Y 2 B Y is formally infiniteg: Let S ¼ fyX g j y 2 R n f0g; g 2 Hg denote the set of nonzero homogeneous elements of R½H: Then S À 1 ðR½HÞ ¼ K½G is called the homogeneous field of quotients of R½H: It is wellknown that K½G is a completely integrally closed t-B ezout domain [2,Theorem 2.2]. An ideal A of R½H is called homogeneous if for all formally infinite ðx g Þ g2H 2 R H such that P g2H x g X g 2 A we have that x g X g 2 A for all g 2 H (equivalently, A is generated by homogeneous elements of R½H). Let I be an ideal of R and let Y be an s-ideal of H. Then I½Y is a homogeneous ideal of R½H: Also note that if J is an ideal of R and Z is an s-ideal of H, then I½YJ½Z ¼ ðIJÞ½Y þ Z: Finally, note that if R is an integral domain, then the t-system on R and the "classical" t-operation on R coincide for nonzero ideals of R. More precisely, the t-system on R extends the t-operation on R to arbitrary subsets of R. For this reason, we do not have to distinguish between the ring theoretical and the monoid theoretical definition of "t" on integral domains. Since the monoid ring R½H is an integral domain (if H is a (torsionless) grading monoid), these considerations also apply to R½H: Lemma 6.1. Let R be an integral domain, H a grading monoid, I an ideal of R, Y an s-ideal of H and A a nonzero ideal of R½H: (4) Let S denote the set of nonzero homogeneous elements of R½H: We only need to show that if A ¼ F t for some nonempty F S; then A ¼ J½Z for some t-ideal J of R and some t-ideal Z of H. Set T ¼ fE S j ; 6 ¼ E A; jEj < 1g: Observe that A ¼ [ E2T E v : Let D 2 T: By [3, Proposition 2.5] we have that D v is a homogeneous divisorial ideal of R½H: It follows from [7, Proposition 2.5] that there exist an ideal J D of R and an s-ideal Z D of H such that D v ¼ J D ½Z D : Therefore, for each C 2 T; there exist an ideal J C of R and an s-ideal Z C of H such that and hence J B J C and Z B Z C (since B v 6 ¼ f0g). Consequently, J is an ideal of R and Y is an s-ideal of H. Moreover, A ¼ [ E2T J E ½Z E ¼ J½Z: (Note that if x 2 J½Z; then x can be represented as a finite sum of elements of the form x b X b with x b 2 J and b 2 Z; and hence there is some E 2 T such that all homogeneous components of x are in J E ½Z E :) We (5) Let R½H be integrally closed and A a t-ideal of R½H that contains a nonzero homogeneous element x 2 R½H: Let f 2 A: Then there is some finite E A such that fx; f g E v : It follows from [4, Theorems 3.2 and 3.7] that E v is homogeneous. Therefore, all homogeneous components of f are contained in E v A: w Proposition 6.2. Let K be a field and G a nontrivial torsionfree Abelian group. The following are equivalent: (1) The radical of every principal ideal of K½G is principal.
(3) K½G is a t-SP-domain. (4) Every nonzero prime t-ideal of K½G contains a nonzero radical principal ideal of K½G: If G satisfies the ascending chain condition on cyclic subgroups, then these equivalent conditions are satisfied.
Proof. The equivalence is an immediate consequence of Theorem 3.10 and Corollary 4.5. Now let G satisfy the ascending chain condition on cyclic subgroup. It follows from [2, Theorem 2.3(a)] that K½G is factorial, and thus K½G is radical factorial. w Note that the equivalent conditions in Proposition 6.2 are not always satisfied. Let p be a prime number, G a nontrivial additive torsionfree p-divisible Abelian group (e.g. ðG; þÞ ¼ ðQ; þÞ or ðG; þÞ ¼ ðZ½ 1 p ; þÞ) and K a field of characteristic p. Then K½G does not satisfy the equivalent conditions in Proposition 6.2. Assume to the contrary that the radical of every principal ideal of K½G is principal. Let a 2 G be nonzero. There is some f 2 K½G such that ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ð1 þ X a ÞK½G p ¼ fK½G: Consequently, there is some m 2 N such that f p m 2 ð1 þ X a ÞK½G: There is some nonzero b 2 G such that p m b ¼ a: Observe that K½G has also characteristic p, and hence 1 þ X a ¼ ð1 þ X b Þ p m : Note that f 1þX b is an element of the field of quotients of K½G: Since K½G is completely integrally closed, and thus root-closed, we infer that f 2 ð1 þ X b ÞK½G: It follows that ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi (1) R½H is integrally closed and every t-ideal A of R½H with A \ S 6 ¼ ; is a finite t-product of radical t-ideals of R½H: (2) R½H is integrally closed and every homogeneous t-ideal of R½H is a finite t-product of radical t-ideals of R½H: (3) Every t-ideal A of R½H with A \ S 6 ¼ ; is a finite t-product of homogeneous radical t-ideals of R½H: (4) Every homogeneous t-ideal of R½H is a finite t-product of homogeneous radical t-ideals of R½H: (5) R is a t-SP-domain and H is a t-SP-monoid.
(1) ) (3), (2) ) (4): This is an immediate consequence of Lemma 6.1(5). (4) ) (5): Let I be a nonzero t-ideal of R and let Y be a nonempty t-ideal of H. Then I½Y is a homogeneous t-ideal of R½H by Lemma 6.1(4). Therefore, there exist n 2 N and finitely many homogeneous radical t-ideals A i of R½H such that I½Y ¼ ð Q n i¼1 A i Þ t : It follows from Lemma 6.1 that, for each j 2 ½1; n; there is some radical t-ideal I j of R and some radical t-ideal Y j of H such that P n i¼1 Y i Þ t ; and hence I ¼ ð Q n i¼1 I i Þ t and Y ¼ ð P n i¼1 Y i Þ t : (5) ) (1): It follows from [19, Proposition 3.10(3)] that R and H are completely integrally closed. Therefore, R½H is completely integrally closed, and hence it is integrally closed. Now let A be a nonzero t-ideal of R½H such that A \ S 6 ¼ ;: By Lemma 6.1 there exist a t-ideal I of R and a t-ideal Y of H such that A ¼ I½Y: There exist n; m 2 N; finitely many radical t-ideals I i of R such that I ¼ ð Q n i¼1 I i Þ t and finitely many radical t-ideals Y j of H such that Y ¼ ð P m j¼1 Y j Þ t : We infer by Lemma 6.1 that I i ½H is a homogeneous radical t-ideal of R½H for all i 2 ½1; n and R½Y j is a homogeneous radical t-ideal of R½H for all j 2 ½1;m: Finally, we have that A ¼ ðI½HR½YÞ t ¼ ðð Q n i¼1 I i Þ t ½HR½ð P m j¼1 Y j Þ t Þ t ¼ ðð Let K be a field, G ¼ Z ðN 0 Þ (i.e., G is isomorphic to the free Abelian group with basis N 0 ) and H ¼ fðx j Þ j2N 0 2 G j x 0 ! x i ! 0 for all i 2 N 0 g: Note that G is isomorphic to the direct sum of countably many copies of Z: Clearly, H is a grading monoid. It follows from [20,Example 4.2] that H is a t-SP-monoid and C t ðHÞ is trivial (since H is t-local). Let ðX i Þ i2N 0 be a sequence of independent indeterminates over K. Set T ¼ K½f Q 1 i¼0 X a i i j ða i Þ i2N 0 2 Hg and S ¼ K½fX i ; X À 1 i j i 2 N 0 g: It is clear that K½H ffi T; T is a subring of K½fX i j i 2 N 0 g and K½G ffi S is factorial. First we show that T is not radical factorial. Let f ¼ X 3 0 ðX 1 þ 1Þ 2 ðX 3 2 þ X 1 Þ: Then f 2 T n T Â : It is sufficient to show that f is an atom of T that is not radical. Since K½X 1 is factorial, it follows by Eisenstein's criterion that X 3 2 þ X 1 is a prime element of K½X 1 ; X 2 : Therefore, X 3 2 þ X 1 is a prime element of K½fX i j i 2 N 0 g: It is clear that X 0 and X 1 þ 1 are prime elements of K½fX i j i 2 N 0 g: Let g; h 2 T be such that f ¼ gh. Since K½fX i j i 2 N 0 g is factorial, there are g 2 K Â ; a 2 f0; 1; 2; 3g; b 2 f0; 1; 2g and c 2 f0; 1g such that g ¼ gX a 0 ðX 1 þ 1Þ b ðX 3 2 þ X 1 Þ c and h ¼ g À 1 X 3 À a 0 ðX 1 þ 1Þ 2 À b ðX 3 2 þ X 1 Þ 1 À c : Without restriction let c ¼ 1. Since g 2 T; we infer that a ¼ 3, and thus b ¼ 2 (since h 2 T). This implies that h ¼ g À 1 2 K Â ¼ T Â ; and hence f is an atom of T. Note that X 1 þ 1 and X 3 2 þ X 1 are prime elements of S. Since S is factorial and f is not a square-free product of prime elements of S, we have that f is not a radical element of S. Since S is a quotient overring of T, we infer that f is not a radical element of T. Consequently, T is not radical factorial. Since H is completely integrally closed, it follows by [7, Lemma 2.1 and Corollary 2.10] that C t ðTÞ ffi C t ðHÞ; and thus C t ðTÞ is trivial. Therefore, if T is a t-SP-domain, then T is radical factorial, a contradiction.
Next we provide a simple way to construct nontrivial examples of w-SP-monoids (or t-SPmonoids) that are grading monoids (if nontrivial examples of w-SP-domains or t-SP-domains are already given). Note that if H is root-closed, then H is a grading monoid if and only if H Â is torsionfree. (If H is a grading monoid, then H is torsionless, and thus H Â is torsionfree. Now let H be root-closed and let H Â be torsionfree. Let n 2 N and x; y 2 H be such that nx ¼ ny. Then nðx À yÞ ¼ 0 2 H; and thus x À y 2 H; since H is root-closed. We infer that x À y 2 H Â : Since H Â is torsionfree and nðx À yÞ ¼ 0; we have that x ¼ y. Therefore, H is a grading monoid.) Remark 6.5. Let R be an integral domain, H a monoid and U a subgroup of H Â with U 6 ¼ H: Set H=U ¼ fxU j x 2 Hg and let V be a subgroup of R Â such that R Â =V is torsionfree (e.g. V ¼ R Â or V ¼ fx 2 R j x n ¼ 1 for some n 2 Ng) and V 6 ¼ R : (1) R is a w-SP-domain (resp. a t-SP-domain) if and only if R is a w-SP-monoid (resp. a t-SP-monoid).
(2) H is a w-SP-monoid (resp. a t-SP-monoid) if and only if H/U is a w-SP-monoid (resp. a t-SP-monoid). (3) R is a w-SP-domain (resp. a t-SP-domain) if and only if R =V is a w-SP-monoid (resp. a t-SP-monoid). If these equivalent conditions are satisfied, then R =V is a grading monoid.

Proof.
(1) Note that f : I t ðRÞ ! I t ðR Þ defined by f ðIÞ ¼ I n f0g for each I 2 I t ðRÞ is a semigroup isomorphism. Moreover, if I 2 I t ðRÞ; then I is radical if and only if f(I) is radical. Therefore, the statement is an immediate consequence of Theorem 4.2 and Corollary 4.4.  (1) and (2). Set A ¼ R =V and suppose A is a t-SPmonoid. Clearly, A is a root-closed monoid whose elements are cancellative. Observe that A Â ¼ R Â =V is torsionfree. Therefore, A is a grading monoid.
w Note that the converse of Remark 6.7 is not true, since there is a factorial domain S for which SvXb is not factorial (as shown in [21]). Clearly, S is a t-B ezout t-SP-domain and a Krull domain. Therefore, SvXb is a Krull domain as well, but it fails to be a t-B ezout domain, since a t-B ezout Krull domain is obviously a factorial domain.