Radical factorization in commutative rings, monoids and multiplicative lattices

In this paper we study the concept of radical factorization in the context of abstract ideal theory in order to obtain a unified approach to the theory of factorization into radical ideals and elements in the literature of commutative rings, monoids and ideal systems. Using this approach we derive new characterizations of classes of rings whose ideals are a product of radical ideals, and we obtain also similar characterizations for classes of ideal systems in monoids and star ideals in integral domains.


Introduction
This article is concerned with the factorization of ideals in commutative rings and monoids into products of radical ideals. Much is known about the integral domains, rings and cancellative monoids whose ideals possess this factorization property; see [1,7,9,14,15,16,18,19] and their references. While by many measures, radical factorization is quite a bit weaker than prime factorization, it is still the case that a ring or monoid whose ideals have radical factorization must meet a number of strong demands, as is evidenced in the characterizations in the cited references. However, one also finds factorization into radical ideals among special subclasses of ideals of rings and monoids. Rather than require all ideals to have the radical factorization property, we thus can consider restricted classes of ideals. This is analogous to the passage from Dedekind domains to Krull domains: The property that every proper ideal of a domain is a product of prime ideals characterizes Dedekind domains, and hence is rather restrictive. Taking a more flexible approach and working up to divisorial closure, we have the familiar property of Krull domains that divisorial ideals factor into prime ideals up to divisorial closure; i.e., every proper divisorial ideal I is of the form I = ((P 1 · · · P n ) −1 ) −1 for some height 1 prime ideals P 1 , . . . , P n . Thus by working with a restricted class of ideals and a more flexible interpretation of product we find Dedekind factorization outside the class of Dedekind domains.
Our goal in this article is to show that the radical factorization property also can be found in more general settings by suitably restricting the ideals considered and having a more flexible notion of product. In fact, our methods allow us to work with both ideal systems of commutative rings as well as monoids. Rather than develop ad hoc approaches to each of these different settings, we give a unified treatment through the use of multiplicative lattices. The collections of ideals that we will be interested in (in both the ring and monoid settings) can be viewed in an obvious way as a lattice having a multiplicative structure. On a more philosophical level, this approach shows that the phenomenon of radical factorization, at least to the extent that we consider it here, is a consequence of the arithmetic of the ideals of the ring, monoid or ideal system, rather than the elements in these ideals, i.e., our analysis of these properties involves quantification over ideals rather than elements. As we recall in Section 2, multiplicative lattices have been well studied by many authors, and so there are a number of tools available for our purposes.
Thus we develop first in Sections 2-6 a theory of radical factorization for multiplicative lattices and use the results obtained in this fashion to derive in Sections 7 and 8 a number of results and characterizations of radical factorization in commutative rings, monoids and ideal systems.
Throughout the paper we assume all rings, monoids and semigroups are commutative and have more than one element.

Definition 2.2.
A multiplicative lattice L is a C-lattice if the set L * of compact elements is multiplicatively closed (i.e., 1 ∈ L * and xy ∈ L * for all x, y ∈ L * ) and every element in L is a join of compact elements. The argument in [10, Lemma 1] shows that a multiplicative lattice L is a C-lattice if and only if there is some multiplicatively closed set A ⊆ L * such that every element of L is a join of elements from A. Notation 2.3. Let L be a multiplicative lattice, and let x, y ∈ L. We use the following notation.
(2) √ x = {y ∈ L | y n ≤ x for some n ∈ N}. An element x is -radical if x = √ x. In the literature on multiplicative lattices, an -radical element is called simply a radical element, but we use the term -radical since there exists the different notion of radical elements in monoids (i.e., an element of a monoid is called radical if the ideal generated by it is a radical ideal). To avoid similar confusion, we use the terms -principal and -invertible in the next definition in place of what are called principal and invertible elements in the context of multiplicative lattices.
One motivation for consideration of multiplicative lattices is that these structures capture fundamental properties of ideals of commutative rings. The notion of a principal element in a multiplicative lattice, first introduced by Dilworth (see [6] for an overview of the history of this notion), plays a role similar to that of finitely generated locally principal ideals in commutative rings (see [3] and [13, Theorem 2] for more on this). Weaker versions of principality also prove useful since they encode familiar properties such as being a multiplicative or cancellative element (see [2]). Definition 2.4. Let L be a multiplicative lattice, and let x ∈ L.
(1) x is cancellative if it is a cancellative element of the monoid (i.e., xy = xz implies y = z for all y, z ∈ L or equivalently xy ≤ xz implies y ≤ z for all y, z ∈ L).
x) for all y, z ∈ L. (6) x is -principal if it is both meet and join principal. (7) x is -invertible if x is -principal and cancellative. The lattice L is principally generated if each element is a join of -principal elements.
Note that in a multiplicative lattice domain, every nonzero -principal element is cancellative and hence is -invertible.
The next lemma collects several useful properties of -principal andinvertible elements; see [4,Lemma 2.3] and [6, Corollary 3.3]. Lemma 2.5. Let L be a multiplicative lattice, and let x, y ∈ L.
(2) xy is -invertible if and only if x and y are -invertible.
As we recall next, the fact that a C-lattice has a good supply of compact elements allows for a localization theory that behaves like that of commutative rings. Notation 2.6. Let L be a C-lattice, and let p ∈ L be -prime (i.e., p = 1 and for all a, b ∈ L, ab ≤ p implies a ≤ p or b ≤ p). We use the terminology of -prime elements to distinguish these elements from the prime elements in monoids. For each x ∈ L, we set Lemma 2.7 (cf. [11, pp. 201-203]). Let L be a C-lattice, let x, y ∈ L and let p ∈ L be an -prime element. ( If x is both weak meet and weak join principal, then x is compact.
Lemma 2.8. Let L be a C-lattice, x ∈ L and p ∈ L an -prime element which is minimal above x.
Since Ω is a multiplicatively closed set of compact elements of L, there is some -prime q ∈ L such that x ≤ q and z ≤ q for each z ∈ Ω. Note that if c ∈ L * is such that c ≤ p, then c ∈ Ω, and hence c ≤ q. Therefore, x ≤ q ≤ p, and hence a ≤ p = q. Since, a ∈ Ω, we have that a ≤ q, a contradiction. Definition 2.9. If L is a multiplicative lattice and the length of the longest chain of -prime elements is n, then the dimension of L is n − 1.
We will be mainly interested in zero-dimensional elements of L, i.e., those elements x for which the only -prime elements above x are maximal.
A lattice L is modular if for all x, y, z ∈ L such that x ≤ z it follows that (x ∨ y) ∧ z = x ∨ (y ∧ z) (equivalently, for all x, y, z ∈ L such that x ≤ z we have that (x ∨ y) ∧ z ≤ x ∨ (y ∧ z). In Section 8, whether a multiplicative lattice is modular is a key issue for determining the ideal systems to which our methods can be applied. The relevance of the modularity condition is due to the following lemma. (1) x is weak join principal.
(2) If x is weak meet principal, then x is meet principal.
(3) If L is modular and x is weak meet principal, then x is -principal.
In particular, if L is modular, then an element of L is -invertible if and only if it is weak meet principal and cancellative. Proof.
(2) Let x be weak meet principal. First we show that yx ∧ zx = (y ∧ z)x for each y, z ∈ L. Let y, z ∈ L. Since x is weak meet principal and yx ∧ zx ≤ x, there is some a ∈ L such that yx ∧ zx = ax. We infer that ax ≤ yx and ax ≤ zx. Therefore, a ≤ y ∧ z, and hence xy ∧ xz ≤ x(y ∧ z) ≤ xy ∧ xz. Now let y, z ∈ L. Then (3) Let L be modular and let x be weak meet principal. By (2) it remains to show that x is join principal. Let y, z ∈ L. Note that yx ≤ x, and hence since x is weak meet principal. Since x is cancellative, we infer that

Radical factorization in C-lattices
The purpose of this section is to give a sufficient condition in Theorem 3.1 for a zero-dimensional element of a C-lattice to factor into a product of -radical elements. An application of this to commutative rings is given in Theorem 7.8. In the next section, we use Theorem 3.1 to find necessary and sufficient conditions for a lattice domain to have the property that every element is a product of -radical elements. Theorem 3.1. Let L be a C-lattice, and let x = 1 be a zero-dimensional element of L. If each maximal element above x is also above a zero-dimensionalradical element that is compact and weak meet principal, then x = y 1 · · · y k for some -radical elements y 1 ≤ · · · ≤ y k .
Proof. We prove the theorem by establishing a series of claims.
By assumption there is a zero-dimensional -radical element y ∈ L such that y ≤ m and y is weak meet principal and compact. Observe that y m = m = √ x m by Lemma 2.8. Next we show that (x ∨ y) n ≥ √ x n for all n ∈ Max(L). Let Using Lemma 2.7(5), we may verify the equality locally by showing that if m ∈ Max(L), Let m ∈ Max(L). Consider first the case that x ≤ m. By assumption, there exists an -radical zero-dimensional element y ∈ L such that y ≤ m and y is compact and weak meet principal. Since y is a weak meet principal element of L, we have that y m is a weak meet principal element of L m . Moreover, it follows by Lemma 2.8 that √ x m = m = y m , and thus √ x m is a weak meet principal element of L m . Therefore, ) m by Claim 1. On the other hand, if m ∈ Max(L) is not above x, then Claim 1 and the assumption that x ≤ m imply Since we have shown this equality holds for all m ∈ Max(L), we conclude by Lemma 2.7(5) Claim 3. There exist a positive integer t and zero-dimensional -radical elements z 1 , . . . , z t such that z 1 · · · z t ≤ x.
Let {m α } denote the collection of maximal elements above x. By assumption, for each α there is a zero-dimensional -radical element y α ∈ L such that y α ≤ m α and y α is compact and weak meet principal. Consider the element a = α ( √ x : y α ). We show a = 1. For each α, since y α is compact we have by Lemma 2.7(6) that Therefore, a mα = 1, which forces a ≤ m α . Since x ≤ a, the maximal elements above a are among the m α . Therefore, a = α a mα = 1.
By Claim 2, x = √ xx 1 for some x 1 ∈ L. If x 1 = 1, then since x 1 is zerodimensional, we may apply Claim 2 to obtain x = √ x √ x 1 x 2 for some x 2 ∈ L.
Continuing in this fashion, we obtain that either x = y 1 · · · y k for someradical elements y 1 ≤ y 2 ≤ · · · ≤ y k , in which case the proof is complete, or there are -radical elements y 1 ≤ y 2 ≤ · · · such that for each i ∈ N, there is Suppose the latter case holds. By Claim 3 there are zero-dimensional -radical elements z 1 , . . . , z t ∈ L such that z 1 · · · z t ≤ x. We claim that x = y 1 y 2 · · · y t+1 .
We verify the equality x = y 1 y 2 · · · y t+1 locally. Let m ∈ Max(L), and suppose first that x t+1 ≤ m. Using the fact that we have that Therefore, x = y 1 · · · y t+1 since this equality holds locally.

Corollary 3.2.
Let L be C-lattice, and let y be a compact zero-dimensionalradical element in L that is weak meet principal. Then for each Proof. Apply Theorem 3.1.

Radical factorization in lattice domains
In this section we characterize the radical factorization property in C-lattices for which every element is a join of -invertible elements. The characterization in Theorem 4.6, the main result of this section, will serve as a basis for most of the applications given in later sections.
The proof of Theorem 4.6 relies on three technical lemmas, all of which are motivated by arguments from [1], although our proofs are more complicated due to the generality of our setting. We show in Section 7 how to derive some of the results from [1] in our context.

Lemma 4.2.
Let L be a C-lattice, and let p be an -prime element of L. If x is a compact join principal element such that x ≤ p and x 2 ∨ p is a product of -radical elements, then for each -prime element q minimal over x ∨ p, we have that p ≤ (q 2 ) q and q = (q 2 ) q .
Proof. Let q be an -prime element minimal over x∨p. By assumption, Since q is minimal over x ∨ p, and hence minimal over

Lemma 4.3. Let L be a C-lattice, and let p < q be -prime elements of L.
Suppose there is a compact weak meet principal element x ≤ q with x q = q. If p is a join of compact weak join principal elements, each of which is a product of -radical elements, then p = 0 q . Proof. Let z be a compact weak join principal element with z ≤ p such that z = z 1 · · · z n , where each z i is an -radical element of L. Since p is a join of such elements, to prove that p = 0 q it suffices to show that z q ≤ 0 q . Now z ≤ p < q = x q , so without loss of generality, (3) L is a Prüfer lattice, i.e., each compact element is -principal.
Proof. (1) The main part of the proof of (1) consists in showing that if p is a nonmaximal -prime element, then p = 0 m for all m ∈ Max(L) above p. For suppose that this has been established and q ≤ p < m are -prime elements and m ∈ Max(L). Then p = 0 m = q by the claim, which in turn implies that dim L ≤ 1. Therefore, we focus in the proof on showing that if p is a nonmaximal -prime element, then p = 0 m for all m ∈ Max(L) above p.
Let p be a nonmaximal -prime element in L, let m ∈ Max(L) with p < m, and let x be an -principal element in L such that x ≤ m and x ≤ p. Let q be an -prime element with q ≤ m and q minimal over p∨x. By Lemma 4.2, q = (q 2 ) q . Let y be an -principal element in L such that y ≤ q and y ≤ (q 2 ) q . Write y = y 1 · · · y k , where the y i are -radical elements. Then y q = ((y 1 ) q · · · (y k ) q ) q . With an aim of applying Lemma 4.3, we show that y q = q.
In light of Lemma 2.8, the preceding decomposition of y and the assumption that y ≤ (q 2 ) q , to prove that y q = q it is enough to show that q is a minimal -prime element above y. Assume that there is some -prime element n with y ≤ n < q. Then there exists an -principal element z ≤ q and z ≤ n.
We infer that ((n ) 2 ) n = (n ) 2 , and thus y ≤ (n ) 2 ≤ (q 2 ) q , a contradiction. Now, applying Lemma 4.3, we have that p = 0 q . If q < m, then we may repeat the preceding argument to show that there is an -prime element n with q < n ≤ m such that q = 0 n . Then p = 0 q = (0 n ) q = q q = q, contrary to the choice of q. Therefore, q = m. We conclude that if m ∈ Max(L) with p < m, we have that p = 0 m . This proves the claim.
(2) Let m ∈ Max(L), and let x ∈ L. Without restriction let x ≤ m. Let p ∈ L be -prime with p ≤ m and p minimal over x m . Suppose first that p < m. Then as observed in the proof of (1), p = 0 m , in which case, Indeed, by (2) the elements in L m are totally ordered with respect to ≤. Since x is compact in L, x m is compact in L m . Since every element of L, hence of L m is a join of -principal elements, it follows that x m is a join of finitely many -principal elements in L m . Since the elements in L m are totally ordered, it follows that x m is -principal in L m .
We want to thank T. Dumitrescu for pointing out (by personal communication) that Lemma 4.4(1) can be proved directly (i.e., not relying on   (3) shows that for each m ∈ Max(L), every element of L m is an -principal element. For more on C-lattices with this property (which are called almost principal element lattices), see [10].
Theorem 4.6. The following are equivalent for a principally generated C-lattice domain L.
(1) L is a radical factorial lattice.
(2) dim L ≤ 1 and each -invertible element is a product of -radical elements.
(3) Each nonzero -prime element is maximal and above an -invertibleradical element.
The -radical of each nonzero compact element is -invertible. (6) Every nonzero compact element is -invertible and the -radical of every compact element is compact.
(3) ⇒ (4) We use Theorem 3.1 to prove this implication. Let 0 = y ∈ L, and let m ∈ Max(L) with y ≤ m. By (3), the -prime elements minimal over y are maximal elements. Thus y is zero-dimensional, and so by Theorem 3.1, y is a product of -radical elements y 1 ≤ y 2 ≤ · · · ≤ y k .
Let p ∈ L be a nonzero -prime element. There is some nonzero x ∈ L * such that x ≤ p. Set y = √ x. Then y is an -invertible -radical element of L and y ≤ p. This proves Claim 1.
Claim 2. If x ∈ L * is nonzero and p ∈ L is an -prime element minimal above x, then p is a minimal nonzero -prime element of L.
Let x ∈ L * be nonzero, p ∈ L an -prime element minimal above x and q ∈ L an -prime element such that 0 < q ≤ p. We have to show that p ≤ q. Observe that L p is a principally generated C-lattice domain and Max(L p ) = {p p }. Moreover, if w ∈ L p is a nonzero compact element, then w = t p for some nonzero compact element t ∈ L, and hence √ w = √ t p = √ t p is an -invertible element of L p (since √ t is an -invertible element of L). Therefore, the -radical of every nonzero compact element of L p is -invertible. Also note that x p is a compact element of L p , p p is an -prime element of L p that is minimal above x p , and q p is an -prime element of L p such that 0 p < q p ≤ p p . Therefore, we can assume without restriction that Max(L) = {p}.
Since p is minimal above x, we have that p = √ x is an -invertible element of L. By Claim 1 there is an -invertible -radical element y ∈ L such that y ≤ q. We have that y ≤ p, and since p is weak meet principal, there is some b ∈ L such that y = pb. Assume that b = 1. Since b ≤ p and p is weak meet principal, there is some d ∈ L such that b = pd. Consequently, (pd) 2 ≤ p 2 d = y, and since y is -radical, we have that pd ≤ y. Therefore, b = pd ≤ y ≤ b. It follows that y = b, and thus y = py. Since y is cancellative, we infer that p = 1, a contradiction. This implies that b = 1, and hence p = y ≤ q, which proves Claim 2. Assume to the contrary that there are some m ∈ Max(L) and -invertible elements x, y ∈ L such that x m and y m are not comparable. Observe that x, y ≤ m. Clearly, x ∨ y is nonzero and compact, and thus z 0 = √ x ∨ y is -invertible and -radical. Since x ≤ z 0 , y ≤ z 0 and z 0 is weak meet principal, there are some v, w ∈ L such that x = vz 0 and y = wz 0 . We have that v and w are -invertible. If v m and w m are comparable, say v m ≤ w m , then Set x 0 = x and y 0 = y. Using the observation before, we can recursively construct a sequence (z i ) i∈N0 of -invertible -radical elements of L and sequences (x i ) i∈N0 and (y i ) i∈N0 of -invertible elements of L such that Note that if i ∈ N 0 , then (x i ) m and (y i ) m are not comparable and, in particular, x i , y i ≤ m and z i ≤ z i+1 ≤ m. Observe that z 0 is compact, and hence there is some k ∈ N such that z k 0 ≤ x 0 ∨ y 0 . We infer that Since z k is nonzero and compact, it follows by Claim 2 that p is a minimal nonzero -prime element of L. In particular, if 0 ≤ i ≤ k, then p is minimal Observe that (z 0 ) p is an -invertible element of L p . Therefore, (z 0 ) p = 1 and z 0 ≤ z k ≤ p, a contradiction. This proves Claim 3. Now let p be an -prime element and m ∈ Max(L) such that 0 < p ≤ m. It is sufficient to show that b ≤ p for all nonzero b ∈ L * such that b ≤ m. Let b ∈ L * be nonzero such that b ≤ m. There is some -prime element q ∈ L that is minimal above b such that q ≤ m. Observe that p = {c m | c ∈ L is -invertible and c ≤ p} and Assume that p and q are not comparable. Then there are some -invertible elements c, d ∈ L such that c m ≤ p, c m ≤ q, d m ≤ p and d m ≤ q. We infer that c m and d m are not comparable, which contradicts Claim 3. Therefore, p and q are comparable. If p ≤ q, then since q is a minimal nonzero -prime element by Claim 2, it follows that p = q. In any case we have that b ≤ q ≤ p.
We will prove in Corollary 6.7 that the decomposition x = x 1 · · · x n in statement (4) of Theorem 4.6 is unique if x is nonzero and the x i are proper. The following remark was communicated to us by T. Dumitrescu.

Remark 4.7.
Let L be a principally generated radical factorial C-lattice domain. Then L is isomorphic to the lattice of ideals of some SP-domain.
Proof. By Lemma 4.4(2), L is locally totally ordered. This implies that L m is a modular lattice for each m ∈ Max(L), and hence L is a modular lattice. Now [2,Theorem 3.4] applies to show that L is isomorphic to the lattice of ideals of some Prüfer domain D. Note that the -radical elements of the lattice of ideals of D are precisely the radical ideals of D. Since L is a radical factorial lattice, we infer that D is an SP-domain.

Remark 4.8.
Note that the "L is a principally generated lattice" condition in Lemma 4.4 and Theorem 4.6 cannot be replaced by the condition that every element of L is a join of (weak) meet principal cancellative elements. We consider the monoid H that is constructed in [17,Example 4.2], and we let L be the set of t-ideals of H. (The definition of t-ideals and the t-system of monoids can be found in [17].) Note that H is an (additively written) cancellative monoid and t is a finitary ideal system on H. By Lemma 8.1 we know that L is a C-lattice. It follows by [17,Example 4.2] that L is a radical factorial lattice such that dim L = 2 and every nonzero element of L is cancellative. In particular, L is a multiplicative lattice domain. Moreover, if I ∈ L, then I = {x + H | x ∈ I} and x + H is a meet principal cancellative element of L for each x ∈ H.

Example: upper semicontinuous functions
The purpose of this section is to give a class of examples of radical factorial lattices arising in a topological context. The importance of this class becomes evident in the next section, where it is shown that all principally generated radical factorial C-lattice domains arise this way. In later sections, we interpret these topological results in the context of rings and monoids.
Recall that for a topological space X, a function f : for all k ∈ N. The compactness of the preimages here implies that such a function takes on only finitely many values. A convenient decomposition of such functions is given in Lemma 5.2(5) based on this observation.
Definition 5.1. Let X be a Hausdorff space. We define U (X) to be the monoid of compactly supported upper semicontinuous functions f : X → N 0 with binary operation given by pointwise addition of functions. We define an order ≤ d on U (X) dual to the usual one by for all x ∈ X. With this order, the zero function 0 is the top element of U (X) and is an additive identity for the monoid (U (X), +). For technical reasons, it will be convenient to introduce a bottom element b to this partially ordered set. We define b : is the partially ordered set with this bottom element appended. Observe that Throughout this section, we denote the characteristic function of a subset A of a set X by 1 A ; i.e., 1 A (x) = 1 if x ∈ A and 1 A (x) = 0 if x ∈ A.

Lemma 5.2. Let X be a Hausdorff space. With the order ≤ d , U b (X) is a multiplicative lattice domain with the following properties.
(1) The meet f ∧ d g and join f ∨ d g are given for all x ∈ X by (

2) If F is a nonempty subset of U (X), then d F exists and is given for all
(3) If F is a subset of U (X) that is bounded below in U (X) with respect to ≤ d , then d F exists and is given by where k 0 < k 1 < · · · < k n are positive integers and C 0 ⊇ C 1 ⊇ · · · ⊇ C n are compact subsets of X.
Proof. That U b (X) is a multiplicative lattice (written additively) follows from (1)-(4), which we will establish below, and the observation that addition (since it is pointwise) commutes with the arbitrary join defined in (2). That the bottom element b of U b (X) is -prime is a consequence of the fact that f + g ∈ U (X) for all f, g ∈ U (X). Thus, once we have established (1)-(4), we have that U b (X) is a multiplicative lattice domain for which the top element of U b (X) is the zero function. ([k, ∞)). As a union of two compact sets, h −1 ([k, ∞)) is also compact. Therefore, h ∈ U (X). It is clear that h is the greatest lower bound of f and g with respect to ≤ d . The proof that the join exists and is as claimed is similar, using instead the fact that the intersection of compact sets is closed, hence compact.
(2) The proof of (2) is a straightforward extension of the argument in (1).
Since f is compactly supported, f is bounded, and so f takes on only finitely many positive values, say k 0 < k 1 < · · · < k n are the positive values of f . For each i, let C i = f −1 ([k i , ∞)). Since f is compactly supported and each k i is positive, each closed set C i is compact. Hence each characteristic function 1 Ci is in U (X) and C 0 ⊇ · · · ⊇ C n . Moreover, Conversely, any function of this form is easily seen to be upper semicontinuous and compactly supported.
Proof. By Lemma 5.2, U b (X) is a multiplicative lattice domain. We claim that 1 A is an -radical element of U b (X) for each compact subset A of X. Let A be a compact subset of X. Then 1 A ∈ U b (X). To see that 1 A is -radical, suppose f ∈ U b (X) and nf ≤ d 1 A for some n ∈ N. Then 1 = 1 A (x) ≤ nf (x) for all x ∈ A. Therefore, for each x ∈ A, f (x) = 0 and hence 1 = 1 A (x) ≤ f (x). It follows that 1 A (x) ≤ f (x) for all x ∈ X and hence f ≤ d 1 A . Thus 1 A is an -radical element of U b (X). Applying Lemma 5.2(5), we obtain that U b (X) is a radical factorial lattice.
Remark 5.4. The proof of Theorem 5.3 shows that for each compact subset A of X, 1 A is an -radical element of U (X). The converse is also true: Suppose g is an -radical element of U (X). If g = 0, then g is the characteristic function of the empty set. Suppose g = 0. Since g is bounded, n = max{g(x) | x ∈ X} exists. Since the values of g are nonnegative integers, to prove g is a characteristic function of a closed set, it suffices to show that n = 1.
Thus n1 A ≤ d g, and since g is -radical, we have that 1 A ≤ d g. But then g(x) ≤ 1 for all x ∈ A. Therefore, n = 1. Moreover, since g is compactly supported, A is a closed subset of a compact set and hence is compact.

Representation of radical factorial lattices
In this section we show that every principally generated radical factorial Clattice domain can be represented as the multiplicative lattice U b (X) of compactly supported upper semicontinuous functions studied in the last section. Using this fact, we show in Corollary 6.6 that the structure of a principally generated radical factorial C-lattice domain is determined entirely by the topology of the space of maximal elements of L. Using this description, we obtain in Corollary 6.7 a uniqueness result for the representation of elements as products of -radical elements in such radical factorial lattices. For the next lemma, recall that a topological space is zero-dimensional if it has a basis of clopen sets.

Lemma 6.2.
If L is a principally generated radical factorial C-lattice domain, then X L is a zero-dimensional Hausdorff space.
Proof. To see that X L is Hausdorff, let m, n ∈ X L . Then m ∨ n = 1, so, since 1 is compact and each of x and y is a join of compact elements, there exist compact elements x ≤ m and y ≤ n such that x ∨ y = 1. Therefore, V (x) and V (y) are disjoint open neighborhoods of m and n, respectively, proving that X L is Hausdorff.
To prove that X L is a zero-dimensional space, it suffices to show that for each nonzero compact element y in L, the set V (y) is a compact open subspace of X L . Let y be a nonzero compact element in L. By Theorem 4.6, √ y is again compact. Thus we may assume without loss of generality that y is -radical. Suppose {y α } is a collection of compact elements in L such that V (y) ⊆ α V (y α ). Then We claim that for each α, (y ∨ y α ) ∨ (y : (y ∨ y α )) = 1. Let m ∈ X L . If y∨y α ≤ m, then since dim L ≤ 1 by Theorem 4.6 and y is -radical and nonzero, we have that y m = (y ∨ y α ) m = m. Since y ∨ y α is compact, Lemma 2.7 (6) implies that (y : (y ∨ y α )) m = (y m : (y ∨ y α ) m ) = (m : m) = 1.
The next lemma introduces a valuation-like map v m : L → N 0 ∪ {∞} for each m ∈ X L . This map is used in Lemma 6.4 to define the functions from X L to N 0 that will be our primary interest in this section. Lemma 6.3. Let L be a principally generated radical factorial C-lattice domain, and let m ∈ X L . For each m ∈ X L , define v m : The following properties hold for all nonzero x, y ∈ L.
Proof. (i) First we show that there is some k ∈ N 0 for which x m = m k . Assume to the contrary that x m = m k for all k ∈ N 0 . Then by Lemma 4.4(2), we have that x m = 0 m . However, L is a principally generated C-lattice domain, and so x m is above an -invertible element y in L. Thus y m = 0 m = (y 2 ) m , so that yb ≤ y 2 for some compact b ≤ m. Since y is -invertible, this implies b ≤ y ≤ m, a contradiction.
It remains to show that for each n ∈ N 0 with x m = m n we have that n = v m (x). Let n ∈ N 0 be such that x m = m n . Then x ≤ m n , and hence n ≤ v m (x). Suppose that n < v m (x). Then x m ≤ m n+1 ≤ m n = x m , and hence m n = m n+1 . By Theorem 4.6, there is a zero-dimensional -invertible -radical element z with z m = m. Thus m n = m n+1 implies (z n ) m = (z n+1 ) m . In this case, there is a compact b ≤ m such that bz n ≤ z n+1 . Since z is -invertible, this implies b ≤ z ≤ m, a contradiction. We infer by (i) that v m (xy) = v m (x) + v m (y). Proof. We prove the second assertion first. Let x be a nonzero compact element of L. Since N 0 is a discrete space, to show that α x is continuous it suffices to prove that for each k ∈ N 0 , {m ∈ X L | v m (x) = k} is an open subset of X L . Let k ∈ N 0 . Using Theorem 4.6, write x = x 1 · · · x s for some -radical elements We observe that the join of any two nonzero -radical elements x, y ∈ L is -radical. Note that if n ∈ X L is such that x ∨ y ≤ n, then n ≥ (x ∨ y) n ≥ x n ∨ y n = n ∨ n = n, and hence {(x ∨ y) m | m ∈ X L , x ∨ y ≤ m} = {m ∈ X L | x ∨ y ≤ m}. Since dim(L) ≤ 1 by Theorem 4.6, we infer from Lemma 2.7 that Moreover, for any two -radical elements y and z in L with y ≤ z, we have that .
Set x s+1 = 1. Using these observations and the assumption that the x i 's form a chain, we see that To prove this set is open in X L , it suffices to show that (x k : x k+1 ) is compact. Since x is compact, we have by Lemma 4.4(3) that x is -invertible, and hence by Lemma 2.5(2) each x i is -invertible. From this it follows that ( The compactness of x k implies then that x k ≤ x k+1 y α1 ∨ · · · ∨ x k+1 y αt for some α 1 , . . . , α t . Using the fact that x k+1 is cancellative, we obtain (x k : x k+1 ) ≤ (x k+1 y α1 ∨ · · · ∨ x k+1 y αt : x k+1 ) = y α1 ∨ · · · ∨ y αt , which proves that (x k : x k+1 ) is compact. This shows that α x is continuous.
Next, suppose that x is an element of X L that is not necessarily compact. We claim that α x is upper semicontinuous. To this end, let k ∈ N 0 . Let A be the set of -invertible elements in L below x. Since L is a principally generated C-lattice domain, we have that x = a∈A a. Therefore, Since each a is compact, α a is a continuous function by what we have previously established. Therefore, the last intersection is an intersection of closed sets. Hence α −1 x ([k, ∞)) is closed, which proves that α x is upper semicontinuous.
To see next that α x is compactly supported, let y be an -invertible element in L with y ≤ x. To prove that x is compactly supported, it suffices to show that {m ∈ X L | y ≤ m} is compact in X L . This is the case by Lemma 6.2, so α x is compactly supported. Theorem 6.5. If L is a principally generated radical factorial C-lattice domain, then L and U b (X L ) are isomorphic as multiplicative lattices.
We show next that for all x, y ∈ L, x ≤ y iff φ(x) ≤ d φ(y). Let x, y ∈ L and m ∈ X L . Without restriction let x, y = 0. If x ≤ y, then , so that x n = n vn(x) ≤ n vn(y) = y n for all n ∈ X L by Lemma 6.3(i), and hence x ≤ y by Lemma 2.7 (5).
It is an immediate consequence of the last statement that φ is injective. Finally, to see that φ is onto, observe first that φ(0) = b and every element of U (X L ) is by Lemma 5.2(5) a linear combination of characteristic functions 1 C , where C is compact in X L . Therefore, we need only show that each such characteristic function is in the image of φ. Let C be a compact subset of X L . Without restriction we can assume that 0 ∈ X L . Since C ⊆ a∈L * \{0} V (a), the fact that C is compact implies there is a finite set A ⊆ L * \{0} such that C ⊆ a∈A V (a). Set y = a∈A a. Then y is a nonzero -radical element of L such that C ⊆ V (y).
Since C is closed in X L , there is a collection {y i } of compact elements in L such that C = i U (y i ). We can assume without restriction that each y i is -radical, since U (y i ) = U ( √ y i ) and √ y i is compact for each i by Theorem 4.6. Since also C ⊆ V (y), we have that C = i (V (y) ∩ U (y i )). Now V (y) ∩ U (y i ) = V ((y : y i )). Let z = i (y : y i ). Then C = V (z) and z is anradical element in L since every element above the zero-dimensional -radical element y is -radical. Thus z m = m for each m ∈ Max(L) above z. Since φ(z)(m) = α z (m) = v m (z), it follows that for each m ∈ X L , we have that m ∈ C if and only if φ(z)(m) = 1. Therefore, φ(z) = 1 C , which proves that φ is onto. Proof. If X L and X L are homeomorphic, then it follows that U b (X L ) ∼ = U b (X L ). Theorem 6.5 then implies L ∼ = L . The converse is straightforward.
Vol. 80 (2019) Radical factorization in commutative rings Page 19 of 29 24 Corollary 6.7. Let L be a principally generated radical factorial C-lattice domain. Then each nonzero x ∈ L can be written uniquely as a product of proper -radical elements x 1 ≤ · · · ≤ x n .
Proof. First observe that by Theorem 4.6, each nonzero element of L has such a representation. Let x = 0 be an element of L. Suppose that x = x 1 · · · x n = y 1 · · · y t , where x 1 ≤ · · · ≤ x n and y 1 ≤ · · · ≤ y t are proper -radical elements. Necessarily x 1 = √ x = y 1 , so that since by Theorem 6.5, Since the mapping φ in Theorem 6.5 is injective, we have that x 2 · · · x n = y 2 · · · y t . Repeating the argument we obtain that n = t and x i = y i for all i = 1, 2, . . . , n.

Applications to commutative rings
We now interpret the results of the last sections in the context of commutative rings by viewing the set consisting of the regular ideals of a ring and the zero ideal of the ring as a principally generated lattice domain. By doing so, we obtain in Theorem 7.4 a characterization of a class of rings whose ideals are a product of radical ideals, followed by similar characterizations for domains in Corollary 7.7. Because it comes at no extra expense of effort, we work more generally in the first two lemmas, where the focus is on a situation in which the role that the total quotient ring plays for regular ideals is replaced with a ring extension. The notion of regularity is replaced with a relativized notion that is flexible enough to cover both subclasses of regular ideals as well as ideals that need not contain a nonzerodivisor.
We recall several relevant definitions from [12]. If R ⊆ T is an extension of commutative rings, an ideal I of R is T -regular if IT = T . For example, if T is the total quotient ring of R, then an ideal is T -regular if and only if it is regular in the usual sense of containing a nonzerodivisor, while if T is the complete ring of quotients of R, then an ideal is T -regular if and only if no nonzero element annihilates it.
Any ring of quotients is tight (see [12, p. 39]).  Proof. (1) and (2). Since the intersection of any two T -regular ideals I and J is T -regular (as it contains IJ), and the sum of any two T -regular ideals is T -regular, we may view L as a sublattice of the complete lattice of all ideals of R. It is clear that the join of an arbitrary subset of L is again in L. If F is a subset of L such that I∈F I is not T -regular, then we set F = 0. Otherwise, we set F = I∈F I. With this definition of arbitrary meets, L is a multiplicative lattice (see [5, pp. 409-410]). Moreover, as discussed in [5, p. 410], the residuation (I : J) is the same whether defined relative to L or the lattice of all ideals of R. It follows that L is a multiplicative lattice. Since the nonzero elements in L are T -regular, L is a lattice domain. To see next that L is a C-lattice, we first verify (2). Let I be a compact element in L. Since IT = T , I is the sum of the finitely generated T -regular ideals of R contained in I. Compactness of I now implies that I is finitely generated. The converse, that a finitely generated T -regular ideal is compact in L, is routine. Since every T -regular ideal of R is the sum of the finitely generated T -regular ideals contained in it, this proves that L is a C-lattice.
(3) Let I be a nonzero -principal element in L. Since I is a T -regular ideal and R ⊆ T is tight, there is a T -invertible ideal A contained in I. Since I is weak meet principal, we have that IJ = A for some T -regular ideal J of R. There is some R-submodule D of T for which AD = R. We infer that I(JD) = R, proving that I is T -invertible. This also implies I is cancellative since if IB ⊆ IC for ideals B, C of R, then (JD)IB ⊆ (JD)IC, so that B ⊆ C. Conversely, if I is a T -invertible ideal of R, it is routine to see that I is -principal in L.
In the next lemma we work under the assumption that every T -regular ideal of R is a sum of T -invertible ideals. This can be viewed as a generalization of the Marot property that requires of a ring that every regular ideal is generated by regular elements. However, the former assumption is quite a bit broader than the Marot property, since for example any Prüfer ring also satisfies it. We point this out again in Remark 7.6. Lemma 7.1 situates the lattice of T -regular ideals in the context of multiplicative lattices. With the lemma, we may apply Theorem 4.6 to obtain a characterization of the radical factorization property for T -regular ideals. If R ⊆ T is an extension of rings, we denote by Max  (1) Every T -regular ideal is a product of radical ideals.
(2) Each T -regular ideal is a product of (unique proper ) radical ideals J 1 ⊆ · · · ⊆ J n . (4) Each T -regular prime ideal is maximal and contains a T -invertible radical ideal. (5) The radical of each finitely generated T -regular ideal is T -invertible. (6) The multiplicative lattice consisting of the T -regular ideals and the zero ideal is isomorphic to U b (Max −1 T (R)). (7) R ⊆ T is a Prüfer extension for which the radical of each finitely generated T -regular ideal is finitely generated.
Proof. By Lemma 7.1, the lattice L consisting of the T -regular ideals and the zero ideal is a C-lattice domain. The assumption that every T -regular ideal is a sum of T -invertible ideals implies that L is principally generated. Thus statements (1)-(7) follow from Lemma 7.1, Theorems 4.6 and 6.5 and Corollary 6.7. Specializing to the case where T is the total quotient ring of R, we obtain the main theorem of this section, which generalizes to a larger class of rings some known characterizations of Marot SP-rings and adds a new one. Specifically, under the more restrictive assumption that R is a Marot N -ring, the equivalence of (1)-(6) is proved by Ahmed and Dumitrescu in [1,Theorem 2.12]. (A Marot ring is an N -ring if for each regular maximal ideal M of R, R M is a discrete rank one Manis valuation ring.) In the theorem, we use Max −1 reg (R) to denote the set of maximal regular ideals of R with respect to the inverse topology. Alternatively, this space can be viewed as Max −1 Q(R) (R). Theorem 7.4. The following are equivalent for a ring R for which every regular ideal is a sum of invertible ideals.
(2) Each regular ideal is a product of (unique proper ) radical ideals J 1 ⊆ · · · ⊆ J n . (3) Every regular prime ideal is maximal and every invertible ideal of R is a product of radical ideals of R. (4) Each regular prime ideal is maximal and contains an invertible radical ideal. (5) The radical of each regular finitely generated ideal is invertible. (6) The multiplicative lattice consisting of the regular ideals and the zero ideal is isomorphic to U b (Max −1 reg (R)). (7) R is a Prüfer ring for which the radical of each finitely generated regular ideal is finitely generated.
Proof. Apply Lemma 7.2 in the case in which T is the total quotient ring of R.
As in Remark 7.3, a Prüfer ring has the property that every regular ideal is a sum of invertible ideals, so we have the following corollary.