U-FACTORIZATION OF IDEALS 1. Introduction This ...

U-FACTORIZATION OF IDEALS J.R. JUETT AND C.P. MOONEY Abstract. We study the factorization of ideals of a commutative ring, in the context of the U-factorization framework introduced by Fletcher. This leads to several “U-factorability” properties weaker than unique U-factorization. We characterize these notions, determine the implications between them, and give several examples to illustrate the differences. For example, we show that a ring is a general ZPI-ring if and only if its monoid of ideals has unique factorization in the sense of Fletcher. We also examine how these “U-factorability” properties behave with respect to several ring-theoretic constructions.

1. Introduction This paper will study the factorization properties of the monoid of ideals of a commutative ring, with a focus on “irredundant” factorizations. In order to place this in the appropriate context, we step back for a moment to give a broad overview of factorization theory in general. Factorization of elements in integral domains has received wide attention, with various properties weaker than unique factorization being identified and studied in detail. (See [7] for an overview.) Factorization theory in commutative rings with zero divisors gets considerably more complex, with multiple competing notions of “associate” elements, each of which leads to a different notion of “irreducible”. An additional layer of complexity is added by the fact that with zero divisors we can have “redundant” factorizations of nonzero nonunits (e.g., if there are nontrivial idempotents), and we have to choose whether or not to allow these “redundant” factorizations. The now-standard system for studying factorization with no “irredundancy” requirement was developed by Anderson and Valdes-Leon [13]; the most common approach to “irredundant” factorization is the “U-factorization” introduced by Fletcher [25]. By analogy with these works, one can formulate frameworks for factorization in commutative monoids, and many of their results and definitions carry over mutatis mutandis. Our paper with Anderson [10] studied factorization of ideals using the setup of [13]. Other notable works along these lines include classic papers by Butts [23] and Gilmer [26], who focused on unique factorization of ideals, and more recent papers by Anderson, Kim, and Park [15] and Hetzel and Lawson [31, 32], who studied some weaker ideal factorization properties. Our present paper will approach the problem from the other point of view, that of U-factorization. Section 2 will give a concise summary of the parts of basic factorization theory that we will later need. We discuss both of the aforementioned frameworks, as well as that of reduced factorization. Along the way, we will be offering a couple improvements on known results (namely Theorems 2.1 and 2.2), which are not only applicable to our study of ideal U-factorization but have some interest in their own right. A solid understanding of the Date: May 28, 2018. 2010 Mathematics Subject Classification. 13A05, 13A15, 13E99, 13F15. Key words and phrases. factorization, U-factorization, ideals, commutative rings, multiplicative ideal theory. 1

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definitions and results of this section is essential to understanding the remainder of the paper, as we will be constantly implicitly referencing this material. In Section 3, we will apply the definitions of Section 2 to the monoid of ideals of a commutative ring. The aspects of the theory related to the setup of [13] are discussed in detail in [10], so here we are just focusing on U-factorization of ideals. We examine different “U-factorability” properties of the monoid of ideals, arriving at complete characterizations for a few of the properties. For example, this leads to an interesting new characterization of general ZPI-rings. We determine (almost all of) the implications between the various properties discussed, and give several examples illustrating the differences between the properties. We examine how these properties behave with respect to localization, direct products, polynomial extensions, power series extensions, and idealizations. Before we begin, let us spend a moment to review some general notation, terminology, and conventions that we will be using throughout the paper. All rings will be commutative with 1 6= 0, modules will be unital, and monoids will be commutative. (We do not require a monoid to have an absorbing element.) We will write monoids with multiplicative notation unless indicated otherwise. To avoid some potential confusion, we will denote images and preimages of functions with square brackets rather than parentheses. Let A be a module over a ring R. The annihilator of A is annR (A) := {x ∈ R | xA = 0}, and forSa ∈ A we abbreviate annR (a) := annR (Ra). The set of zero divisors of A is ZR (A) := 06=a∈A annR (a), and the set of regular elements of R is R \ ZR (R). An ideal is regular if it contains a regular element. We denote the set of R-submodules of A by SR (A). For B, C ∈ SR (A), we define (B :R C) := {x ∈ R | xC ⊆ B}. We say R (resp., A) locally satisfies a property P if RM (resp., AM ) satisfies P as a ring (resp., an RM -module) for each maximal ideal M . In the above notation, we will frequently be dropping the subscript “R” when there is no danger of confusion. This same convention will apply to all future notation. We use U (H) to denote the group of units of a monoid H, and we abbreviate H ∗ := H \{0}, H # := H ∗ \ U (H), and H0# := H \ U (H). We denote the set of maximal (resp., of prime, of) ideals of a ring R by Max(R) (resp., Spec(R), I(R)). We call R quasilocal if |Max(R)| = 1, semi-quasilocal if |Max(R)| < ∞, and (semi-)local if it is Noetherian and (semi-)quasilocal. We say R has finite character if every nonzero element is contained in only finitely many maximal ideals. We respectively use J (R) and Nil(R) to denote the Jacobson radical and nilradical of R. We say R is reduced if Nil(R) = (0). A ring is connected or indecomposable if it has no nontrivial idempotents, or equivalently cannot be written as a direct product of two rings, and is completely decomposable if it is a finite direct product of connected rings. A ring is B´ezout if every finitely generated ideal is principal. A chained ring is a quasilocal B´ezout ring, or equivalently a ring with every ideal divided (i.e., comparable to every ideal). A ring is arithmetical if it is locally chained. A valuation (resp., Pr¨ ufer) domain is a chained (resp., arithmetical) domain. A discrete valuation ring (DVR) is a Noetherian valuation domain, or equivalently a local principal ideal domain (PID). A Noetherian Pr¨ ufer domain is called Dedekind, and a locally Dedekind domain is almost Dedekind. Lastly, a special principal ideal ring (SPIR) is an Artinian local principal ideal ring (PIR). 2. Review of Factorization The purpose of this section is to review some of the terminology and results from basic factorization theory. We will give the definitions in a fairly high degree of generality, and

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then apply them later to the monoid of ideals of a ring. In this way, we can place our later study of (U-)factorization of ideals in the appropriate context. Almost all of the definitions presented in this section are by analogy with those in [1, 13, 17, 18]. These papers formulate their work in ring-theoretic terms, but most of the definitions and results carry over mutatis mutandis to a more general monoid setting. In such cases, we will simply point the reader to the appropriate proofs from the literature rather than rehash old arguments. Throughout the rest of the section, let H be a monoid. We call a, b ∈ H associates if (a) = (b). We call b a (proper) divisor of a if (b) (properly) contains (a). We say a ∈ H is cancellative if xa = ya ⇒ x = y, and pr´esimplifiable if a = xa ⇒ x ∈ U (H). We call H pr´esimplifiable (resp., cancellative) if all of its nonzero elements have this property. (The term “pr´esimplifiable” was coined by Bouvier, who studied rings and monoids with this property in several papers, e.g., [19, 20, 21, 22].) Cancellative ⇒ pr´esimplifiable, but not conversely, e.g., a quasilocal ring that is not a domain. A factorization of x ∈ H0# of length n is an expression x = x1 · · · xn with each factor xi ∈ H0# . We call factorizations x1 · · · xm and y1 · · · yn isomorphic if m = n and each (xi ) = (yi ) after a suitable reordering. The factorization length of x ∈ H0# is the supremum L(x) of the lengths of its factorizations, and L(u) := 0 for u ∈ U (H). (This is the classical definition of L. If H satisfies the ascending chain condition on principal ideals, then one can extend L to an ordinal-valued function with a recursive definition, as in the work of Anderson and Juett [8, 9]. In this paper we will always use L to denote classical factorization length rather than recursive factorization length.) We call x ∈ H0# irreducible or an atom if (x) = (y)(z) ⇒ (x) = (y) or (z). Note that prime ⇒ irreducible, and the two notions are equivalent for 0 (c.f. [13, Theorem 2.13]). A factorization with irreducible factors is called atomic. We call H atomic if every nonzero nonunit has an atomic factorization, a bounded factorization monoid (BFM) if L(x) < ∞ for each x ∈ H ∗ , an irreducible divisor finite monoid (idf-monoid) if each nonzero nonunit has only finitely many irreducible divisors up to associates, a weak finite factorization monoid (WFFM) if each nonzero nonunit has only finitely many divisors up to associates, a finite factorization monoid (FFM) if every nonzero nonunit has only finitely many factorizations up to isomorphism, a half factorization monoid (HFM) if it is atomic and any two atomic factorizations of the same nonzero nonunit have the same length, and a unique factorization monoid (UFM) if each nonzero nonunit has a unique atomic factorization up to isomorphism. The notions of FFM, WFFM, and atomic idf-monoid are equivalent for pr´esimplifiable monoids (c.f. [13, Proposition 6.6]). If we wish to specify that H is a ring (resp., domain), we replace the “monoid” with “ring” (resp., “domain”) in the above names and adjust the acronym accordingly. Similar conventions will apply to future definitions. The definitions we have discussed so far are those of the framework developed (in a ringtheoretic context) by Anderson and Valdes-Leon [13]. This remains the most common setup for studying factorization in rings with zero divisors (or more generally non-cancellative monoids), but sometimes one might wish to only consider factorizations that are “irredundant” in some sense. One way to do this is with the (strongly) reduced factorizations recently studied by Anderson, Chun, and Valdes-Leon [24]. By analogy with their ring-theoretic definitions, we call a factorization x = x1 · · · xn reduced ifQx 6= x1 · · · xi−1 xi+1 · · · xn for i = 1, . . . , n. We call the factorization strongly reduced if x 6= i∈Λ xi for any Λ ( {1, . . . , n}. Strongly reduced ⇒ reduced, but not conversely, e.g., the factorization (1, 0) = (2, 0)2 ( 21 , 0)2

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in Q × Q [24, Example 2.3(a)]. One can define and study (strongly) reduced factorization versions of the definitions in the previous paragraph – see [24] for a more detailed discussion. More attention has been given to Fletcher’s older approach [25], which has a different “irredundancy” requirement than (strongly) reduced factorization. In analogy with his (ringtheoretic) definitions, we define a U-factorization of x ∈ H0# of essential length n to be a factorization of the form x = y1 · · · ym z1 · · · zn (m ≥ 0, n ≥ 1) with (x) = (z1 · · · zn ) but (x) 6= (z1 · · · zi−1 zi+1 · · · zn ) for i = 1, . . . , n. We call the yi ’s the inessential factors and the zj ’s the essential factors, and we write x = y1 · · · ym dz1 · · · zn e to indicate which factors are essential. The essential divisors of x are the essential factors in its U-factorizations. We call two U-factorizations y1 · · · ym dz1 · · · zn e and b1 · · · bs dc1 · · · ct e U-isomorphic if the factorizations z1 · · · zn and c1 · · · ct are isomorphic. For x ∈ H0# , we define LU (x) to be the supremum of the essential lengths of its U-factorizations, and LU (u) := 0 for u ∈ U (H). Note that each factorization of a pr´esimplifiable nonunit is already a U-factorization with every factor essential, hence a strongly reduced factorization. Therefore LU (x) = L(x) if x is pr´esimplifiable. Also observe that x ∈ H is irreducible if and only if LU (x) = 1. A U-atomic factorization is a U-factorization with irreducible essential divisors. Fletcher [25] focused his attention on unique U-atomic factorizations. Non-unique U-atomic factorizations were later considered in more depth by A¯garg¨ un, Anderson, and Valdes-Leon [1]; Axtell [17]; Axtell, Forman, Roersma, and Stickles [18]; and Mooney [36]. In analogy with their (ring-theoretic) definitions, we call H U-atomic if every nonzero nonunit has a U-atomic factorization, a U-BFM if LU (x) < ∞ for each nonzero nonunit x, a U-idf-monoid if each nonzero nonunit has only finitely many irreducible essential divisors up to associates, a UFFM if every nonzero nonunit has only finitely many U-factorizations up to U-isomorphism, a U-HFM if it is U-atomic and any two U-atomic factorizations of the same nonzero nonunit have the same essential length, and a U-UFM if each nonzero nonunit has a unique U-atomic factorization up to U-isomorphism. Because H is a U-FFM if and only if every nonzero nonunit has only finitely many essential divisors up to associates (c.f. [17, Theorem 2.9] or Theorem 2.1 below), there is no need to define a “U-WFFM”. Note that, for pr´esimplifiable monoids, the “U-” definitions in this paragraph are each equivalent to the earlier definitions without the “U-”. When formulating the definitions of UFM, HFM, etc., it is clear that we do not want to place any restrictions on the factorizations of 0. However, in the case of U-factorization, it is not clear whether or not we want to place restrictions on the U-factorizations of 0. The existing literature has made no consistent choice on this, so we will give separate names for the two possibilities. In honor of Fletcher’s work [25], we define Fletcher UFM, Fletcher HFM, etc., by removing the “nonzero” from the corresponding “U-” definition. (Fletcher’s original definition of a “unique factorization ring” is that every nonunit has a unique [up to Uisomorphism] U-factorization with every factor irreducible. This turns out to be equivalent to the way we defined Fletcher UFR [13, Theorem 4.9].) Of course, Fletcher atomic is equivalent to U-atomic, but we will see in Example 3.29 below that the other “Fletcher” definitions are strictly stronger than their “U-” counterparts. Again, we must emphasize that some care must be taken when reading the existing literature, due to the fact that sometimes “U-UFR” is used as we use it and sometimes it is used to mean Fletcher UFR, etc. (There are a couple minor errors caused by this inconsistency in otherwise excellent resources. For example, our definition of U-UFR is the same as in [18], and the assertion

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that UFR ⇒ U-UFR [18] is only true with this interpretation, but [18, Theorem 2.1] and [18, Theorem 3.3] are only true if “U-UFR” is interpreted as “Fletcher UFR”.) We end the section with two new results, providing characterizations of Fletcher FFRs and non-connected U-FFRs. These will be a useful stepping stone to understanding the corresponding ideal factorization properties. Theorem 2.1. The following are equivalent for a ring R. (1) R is a Fletcher FFR. (2) Every nonunit has only finitely many essential divisors up to associates. (3) R is an FFD or a finite direct product of SPIRs and finite local rings. Proof. (The equivalence of (1) and (2) can be shown by making small adjustments to the proof of [17, Theorem 2.9], but weQwill present a simple alternative proof.) (1) ⇒ (3): Assume R is a Fletcher FFR. Then R = ni=1 Ri with each Ri connected, for otherwise LU (0) = ∞. It is easy to see that each Ri is a Fletcher FFR. If n ≥ 2 and some Ri has a nonunit regular element s, then {(0, . . . , 0, sk , 0, . . . , 0)}∞ k=1 is an infinite sequence of non-associate essential divisors of (0, . . . , 0), a contradiction. Therefore either n = 1 or each Ri without nonzero zero divisors is a field. So we may assume that R is connected but not a domain. There are only finitely many non-associate powers of each z ∈ Z(R), so by connectedness we have (z)k = (z)2k = (0) for large enough k. Thus Z(R) ⊆ Nil(R) and R is pr´esimplifiable (see Theorem 3.1(2) below). If R has a regular nonunit s, then for each 0 6= z ∈ Z(R) we esimplifiability) zero divisors, a have an infinite sequence {sk z}∞ k=1 of non-associate (by pr´ contradiction. So R is a total quotient ring and therefore has only finitely many (principal) ideals, hence is an SPIR or a finite local ring [38, Proposition 1]. (3) ⇒ (2): Clear. (2) ⇒ (1): It suffices to show that (2) implies R is a Fletcher BFR. Pick x ∈ R0# and let N be the number of non-associate essential divisors of x. If x = y1 · · · ym dz1 · · · zn e is a U-factorization, then z1 , z1 z2 , . . . , z1 · · · zn are non-associate essential divisors of x, hence n ≤ N .  The next result is a slight extension of [17, Theorem 3.10]. Theorem 2.2. A non-connected ring is a U-FFR if and only if it is (i) a finite direct product of SPIRs and finite local rings or (ii) a direct product of an FFD and a field. Proof. Let R = R1 × R2 be a non-connected ring. (⇒): Assume R is a U-FFR. Then each Ri is a Fletcher FFR, so by Theorem 2.1 we may assume R1 is an FFD but not a field. Suppose that R2 is not a field, say x ∈ R2# and y ∈ R1# . Then {(y n , x)}∞ n=1 is an infinite sequence of non-associate essential divisors of (0, x), a contradiction. (⇐): [17, Theorem 3.10].  3. U-factorization in the Monoid of Ideals This section will study the theory of U-factorization in the context of the (multiplicative) monoid of ideals of a ring. This is in contrast to our work with Anderson [10], which focused on factorization rather than U-factorization. Inevitably there will be some small amount of overlap between the two papers (which we have marked with citations), but for the most part the two works are independent of each other. Let R be a ring. Following Ali [3, 4], we call an ideal half cancellation if it is pr´esimplifiable as an element of (I(R), ·). We say R satisfies the half cancellation law (HCL) if (I(R), ·) is pr´esimplifiable. We summarize some basic facts about pr´esimplifiability and half cancellation in the following theorem. For further details, see [11].

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Theorem 3.1. Let R be a ring. (1) An ideal is half cancellation ⇔ it is locally half cancellation [11, Theorem 4.8]. So R satisfies HCL ⇔ it is pr´esimplifiable and locally satisfies HCL [11, Theorem 6.18]. (2) If I is a finitely generated ideal, then it is half cancellation ⇔ ann(I) ⊆ J (R) [11, Corollary 4.10]. Hence (i) an element x ∈ R is pr´esimplifiable ⇔ Rx is half cancellation ⇔ ann(x) ⊆ J (R) [11, Corollary 4.6(3)], and (ii) R is pr´esimplifiable ⇔ Z(R) ⊆ J (R) ⇔ nonzero finitely generated ideals of R are half cancellation ⇔ nonzero (principal) ideals of R are locally nonzero [11, Theorem 6.9]. We call an ideal of a ring R weakly nonfactorable if it is irreducible as an element of (I(R), ·). Similarly, we say R is weakly factorable if (I(R), ·) is atomic. To explain the “weakly” terminology, a proper ideal I is called nonfactorable if I = I1 I2 ⇒ either (i) some Ii = R or (ii) some Ii = I = (0). (This definition is essentially due to Butts [23], with the minor modification that we allow (0) to be nonfactorable, to be consistent with the factorization framework of Anderson and Valdes-Leon [13].) Note that an ideal I is nonfactorable if and only if L(I) = 1 or I = (0) is prime. Anderson, Kim, and Park [15], and later Hetzel and Lawson [32], studied factorable rings, which are rings where every proper ideal is a product of nonfactorable ideals. We call R a bounded ideal factorization ring (BIFR), (resp., weakly nonfactorable divisor finite ring (wndf-ring), weak finite ideal factorization ring (WFIFR), finite ideal factorization ring (FIFR), half ideal factorization ring (HIFR), unique ideal factorization ring (UIFR)) if (I(R), ·) is a BFM, (resp., idf-monoid, WFFM, FFM, HFM, UFM). We define U-UIFR, Fletcher UIFR, etc., in the obvious analogous way. Inspired by Alan’s “finite decomposition rings” [2, Definition 19], we call a ring a U- (resp., Fletcher) finite ideal decomposition ring (FIDR) if it is weakly factorable and every nonzero (resp., every) proper ideal has only finitely many U-atomic factorizations. The following theorem shows that there is a close relationship between U-factorizations and reduced factorizations of ideals. In view of these observations, we will tend to work with reduced factorizations of ideals rather than U-factorizations, for the sake of simplicity. In this way, the theory of U-factorization of ideals works out a bit more cleanly than that of elements. For example, in Theorem 3.3 we will see that atomic and U-atomic are equivalent for monoids of ideals. By contrast, whether these properties are equivalent for rings has remained an open problem for over twenty years [13, Question 3.12]. Theorem 3.2. Let R be a ring. (1) Reduced factorizations in I(R) are in fact strongly reduced. (2) If J1 · · · Jm dI1 · · · In e is a U-factorization of I ∈ I(R)# 0 , then I = I1 · · · In is a reduced factorization. Therefore LU (I) is the supremum of the lengths of the reduced factorizations of I. (3) If R is weakly factorable and I ∈ I(R)# 0 , then LU (I) is the supremum of the lengths of the reduced atomic factorizations of I. Proof. Q Q (1) If I = I1 · · · In = i∈Λ Ii for some Λ ( {1, . . . , n}, then I = i6=k Ii for any k ∈ / Λ. (2) Follows immediately from the fact that the associate relation in I(R) is equality. (3) Assume R is weakly factorable. We need to show that for any reduced factorization I = I1 · · · In , there is a reduced atomic factorization of I of length ≥ n. Qm i Let Ii = j=1 Ji,j be an atomic factorization. Remove redundant factors from

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Q Q i Q I = ni=1 m j=1 Ji,j until we arrive at a reduced atomic factorization I = (i,j)∈Λ Ji,j . For each i = 1, . . . , n, we must have at least one (i, j) ∈ Λ, for otherwise I = I1 · · · Ii−1 Ii+1 · · · In . Therefore |Λ| ≥ n, as desired.  The following two results show that defining terms like “weakly U-factorable”, “U-WFIFR”, or “Fletcher WFIFR” would be redundant. Theorem 3.3. A ring is weakly factorable ⇔ every (nonzero) proper ideal has a reduced atomic factorization ⇔ every (nonzero) proper ideal has a U-atomic factorization. Proof. The first equivalence is clear, and the second follows from Theorem 3.2(2).



Theorem 3.4. The following are equivalent for a ring R. (1) The ring R is a U- (resp., Fletcher) FIFR. (2) Each nonzero (resp., Each) proper ideal has only finitely many reduced factorizations. (3) Each nonzero (resp., Each) proper ideal has only finitely many essential divisors. Proof. (1) ⇔ (2): Theorem 3.2(2). (1) ⇔ (3): Adjust the proof of Theorem 2.1.



The following theorem generalizes the fact that a ring satisfying HCL is an FIFR if and only if it is a weakly factorable wndf-ring [10]. Theorem 3.5. The following are equivalent for a ring R. (1) The ring R is a U- (resp., Fletcher) FIDR. (2) The ring R is weakly factorable and every nonzero (resp., every) proper ideal has only finitely many reduced atomic factorizations. (3) The ring R is a weakly factorable U- (resp., Fletcher) wndf-ring. Proof. (1) ⇔ (2): Theorem 3.2(2). (2) ⇒ (3): By contrapositive. Assume that R is weakly factorable but there is a nonzero (resp., a) proper ideal I with infinitely many weakly nonfactorable essential divisors. It suffices to show that each weakly nonfactorable essential proper divisor J of I appears in a reduced atomic factorization of I. Say I = JA is a reduced factorization. Let A = A1 · · · An be an atomic factorization. Remove redundant factors from I = JA1 · · · An to obtain a reduced atomic factorization of I. Since I ( A, this new factorization must still have J as a factor. (3) ⇒ (2): Let I be a nonzero (resp., a) proper ideal of a weakly factorable U- (resp., Fletcher) wndf-ring. Each reduced atomic factorization of I has (up nm to order) the form I = J1n1 · · · Jm (ni ≥ 0), where J1 , . . . , Jm are the weakly nonfactorable n1 nm essential divisors of I. Give X := {(n1 , . . . , nm ) ∈ Nm 0 | I = J1 · · · Jm is reduced} the n0 n0 nm product partial order. Since we cannot have reduced factorizations J1n1 · · · Jm = J1 1 · · · Jmm with (n1 , . . . , nm ) < (n01 , . . . , n0m ), the set X is an antichain, hence is finite by Dickson’s Lemma.  The theory of U-factorization has some connection to that of comaximal factorization. In the terminology of McAdam and Swan [35, Section 5], a comaximal factorization is a factorization with pairwise comaximal factors, a proper ideal is pseudo-irreducible if it has no comaximal factorizations of length ≥ 2, and a comaximal factorization is complete if its factors are pseudo-irreducible. Note that comaximal factorizations are reduced, hence weakly nonfactorable ideals are pseudo-irreducible. Hedayat and Rostami recently proved that a ring is J-Noetherian if and only if every (nonzero) proper ideal has a complete comaximal factorization [30, Theorem 2.6]. (Recall that an ideal is J-radical if it is an intersection of maximal ideals. A ring R is J-Noetherian if it satisfies the ascending chain condition on

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J-radical ideals, or equivalently the Zariski topology on Max(R) is Noetherian.) For more information on comaximal factorization of ideals, see [29, 30, 33, 35]. Figure 1 illustrates the relationship between the various “U-factorability” properties. The corresponding diagram for the “Fletcher” properties also holds. Figure 1. Implications Between U-Factorability Properties +3

U-UIFR

U-HIFR

factorable

U-FIFR (



+3

U-FIDR DL weakly factorable



U-BIFR U-FIDR



U-wndf-ring

+3



J-Noetherian ks

%-

 

weakly factorable

Theorem 3.6. The implications in Figure 1 hold, as well as the analogous implications for the “Fletcher” definitions. Proof. We prove the “U-” case. The “Fletcher” case is similar. (U-UIFR) ⇒ (U-HIFR) ⇒ (weakly factorable) ⇐ (factorable): Clear. (U-UIFR) ⇒ (U-FIDR): Clear. (U-FIDR) ⇒ (UBIFR) ⇐ (U-HIFR): Theorem 3.2(3). (U-FIDR) ⇔ (weakly factorable U-wndf-ring): Theorem 3.5. (U-FIFR) ⇒ (U-FIDR): It suffices to show that a U-FIFR R is weakly factorable. Each nonzero proper ideal I has a reduced factorization I = I1 · · · In that is maximal in the sense that n = LU (I) and I has no reduced factorization I = J1 · · · Jn with each Ii ⊆ Ji and at least one inclusion proper. (For otherwise I has infinitely many reduced factorizations.) To see that each Ii is weakly Q nonfactorable, write Ii = AB with Q A, B 6= R.QBy maximality, the factorization I = AB j6=i Ij is not reduced, hence I = A j6=i Ij or B j6=i Ij , hence Ii = A or B. (Weakly factorable) ⇒ (J-Noetherian) ⇐ (U-BIFR): Let R be a weakly factorable ring (resp., a U-BIFR). Then every nonzero proper ideal is a product of pseudo-irreducible ideals (resp., has a finite upper bound on the lengths of its comaximal factorizations). By [33, Theorem 4.11], every nonzero proper ideal has a complete comaximal factorization, or equivalently R is J-Noetherian [30, Theorem 2.6]. (U-wndf-ring) ⇒ (J-Noetherian): By contrapositive. Let R be a ring with an infinite ascending chain J1 ( J2 ( · · · of nonzero J-radical ideals.TFor k ≥ 1, pick a maximal ideal Mk containing Jk but not Jk+1 T∞ T.∞For m < n, we have Mm + ∞ M ⊇ M + J = R. Therefore M = M · · · M ( k m n k 1 n k=n k=1 k=n+1 Mk ) is ∞ a comaximal factorization for each n. So {MkT }k=1 is an infinite set of weakly nonfactorable essential divisors of the nonzero proper ideal ∞  k=1 Mk . Question 3.7. Are all U-BIFRs weakly factorable? (Given that it is still unknown whether U-BFRs are U-atomic [1], we expect this question to be somewhat difficult.) We will see in Example 3.29 below that Figure 1 and its “Fletcher” analog are complete, except possibly for the unresolved issue of Question 3.7. We will need to continue our investigation of U-factorization a little deeper before we are ready to understand the examples.

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Finding a negative answer to Question 3.7 would involve developing some more examples of U-BIFRs. Our main such example is a strongly Laskerian ring. (Recall that a primary ideal is strongly primary if it contains a power of its radical, and a ring is (strongly) Laskerian if every proper ideal is a finite intersection of (strongly) primary ideals.) In general, Noetherian ⇒ strongly Laskerian ⇒ Laskerian ⇒ J-Noetherian, but none of the implications reverse [28]. Laskerian rings need not be weakly factorable (e.g., a one-dimensional non-discrete valuation domain – see Theorem 3.20(1)), Noetherian rings are weakly factorable (c.f. [13, Theorem 3.2]), and we do not know whether strongly Laskerian rings are weakly factorable. Theorem 3.8. Strongly Laskerian rings are Fletcher BIFRs. Proof. (We adapt the proof of [1, Lemma 4.16].) Let R be a strongly Laskerian ring and I be a proper ideal. Write I = Q1 ∩ · · · ∩ Qn with each QQ Pi -primary, say Pisi ⊆ Qi . Let i strongly Q I = I1 · · · Ik be a reduced factorization. Because ( Ij ⊆Pi Ij )( Ij *Pi Ij ) = I ⊆ Qi , we have Q Q length si will Ij ⊆Pi Ij ⊆ Qi . If Ij ⊆Pi Ij has more than si factors, then any subproduct of Q si be contained in Pi ⊆ Qi . So there is a Λi ⊆ {1, . . . , k} with |Λi | ≤ si and j∈Λi Ij ⊆ Qi . P Q S  Thus I ⊆ j∈Sn Λi Ij ⊆ Q1 ∩ · · · ∩ Qn = I, so k = | ni=1 Λi | ≤ ni=1 si . i=1

Question 3.9. Are all strongly Laskerian rings weakly factorable? (c.f. Question 3.7.) Before moving on to other topics, we present two more results about U-factorization length. Theorem 3.10. Let x be a nonzero element of a ring R. We have LU (x) ≤ LU ((x)), with equality if R is quasilocal or every divisor of (x) is principal. Proof. Note that (x) = (x1 ) · · · (xn ) is a reduced factorization if and only if x has a Ufactorization of the form x = y1 · · · ym dx1 · · · xn e. Hence LU (x) ≤ LU ((x)), and equality holds if divisors of (x) are principal. Now assume R is quasilocal and let (x) = I1 · · · In be Q P Q a reduced factorization. Then x = kj=1 ni=1 xi,j with xi,j ∈ Ii . Writing ni=1 xi,j = xyj , P P we obtain x = x kj=1 yj , hence kj=1 yj ∈ U (R) by pr´esimplifiability. Since R is quasilocal, some yj ∈ U (R), so (x) = (x1,j · · · xn,j ). Therefore LU (x) = L(x) ≥ n, as desired.  P Theorem 3.11. Let I = I1 · · · In be a comaximal factorization. Then LU (I) = ni=1 LU (Ii ). Proof. Via induction, weQ reduce LU (I1 ) + LU (I2 ) ≤ LU (I), it Qmtoi the case n = 2. To see that Qm 2 i suffices to show that I = i=1 j=1 Ji,j is reduced if each Ii = j=1 Ji,j is reduced. If not, say Qm 1 Qm1 Qm1 I = ( j=2 J1,j )I2 , then j=2 J1,j = ( j=2 J1,j )(I1 +I2 ) ⊆ I1 +I = I1 , a contradiction. For the reverse inequality, let I = J1 · · · Jk be a factorization with k > LU (I1 ) + LU (I2 ). We compute Q Q Q Q I1 = I1 ( kj=1 (I1 + Jj ) + I2 ) ⊆ kj=1 (I1 + Jj ) + I1 I2 = kj=1 (I1 + Jj ) ⊆ I1 + kj=1 Jj = I1 , Q and similarly I2 = kj=1 (I2 + Jj ). Reduce these expressions for I1 and I2 to a combined total of less than k factors. Q There is some j such that the jth factor was removed from both expressions. Thus I ⊆ l6=j Jl ⊆ I1 ∩ I2 = I and I = J1 · · · Jk is not reduced.  By Theorem 3.6, a ring satisfying any form of “U-factorability” is J-Noetherian, hence completely decomposable. Therefore it is very useful to understand how these properties behave with respect to direct products. Q Lemma 3.12. Let R = ni=1 Ri be a direct product of rings (2 ≤ n < ∞). Q P P (1) For I = ni=1 Ii ∈ I(R), we have L(I) = ni=1 L(Ii ) and LU (I) = ni=1 LU (Ii ).

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Q (2) An ideal ni=1 Ii ∈ I(R) is weakly (resp., is) nonfactorable if and only if some Ij is weakly (resp., is nonzero and) nonfactorable and Ii = Ri for i 6= j [10]. Proof. For (1), c.f. [9, Propositions 4.2 and 4.9(1)]. Part (2) follows from (1) and the fact that an ideal of R is (weakly) nonfactorable ⇔ its (essential) factorization length is 1.  Q Theorem 3.13. Let R = ni=1 Ri be a direct product of rings (2 ≤ n < ∞). (1) R is weakly factorable ⇔ each Ri is weakly factorable [10]. (2) R is factorable ⇔ each Ri is a factorable ring with zero divisors [10]. (3) R is a U- (resp., Fletcher) BIFR ⇔ each Ri is a Fletcher BIFR. (4) R is a U- (resp., Fletcher) wndf-ring ⇔ each Ri is a Fletcher wndf-ring. (5) R is a U- (resp., Fletcher) FIDR ⇔ each Ri is a Fletcher FIDR. (6) R is a U- (resp., Fletcher) HIFR ⇔ each Ri is a Fletcher HIFR. (7) R is a U- (resp., Fletcher) UIFR ⇔ each Ri is a Fletcher UIFR. Proof. Part (3) follows immediately from Lemma 3.12(1), and the rest follows routinely from Lemma 3.12(2).  The interaction between the “FIFR” properties and direct products is more complex. Theorem 3.14. A ring is a Fletcher FIFR if and only if it is a U-FIFD or a finite direct product of SPIRs and finite local rings. Proof. (⇒): Theorem 2.1. (⇐): Since (0) is nonfactorable in a domain, a U-FIFD is a Fletcher FIFR. A finite direct product of SPIRs and finite local rings has only finitely many ideals, hence is a Fletcher FIFR by Theorem 3.4.  Theorem 3.15. A non-connected ring is a U-FIFR if and only if it is (i) a finite direct product of SPIRs and finite local rings or (ii) a direct product of a WFIFD and a field. Proof. (⇒): By Theorem 2.2, we only need to consider a U-FIFR of the form D × K, where D is a domain and K is a field. To see that D is a WFIFD, let I be a nonzero proper ideal. If I1 is a proper divisor of I, say I = I1 I2 , then I × (0) = (I1 × (0))(I2 × K) is a reduced factorization. Therefore I has only finitely many divisors. (⇐): By Theorem 3.14, we only need to consider a ring of the form D × K, where D is a WFIFD and K is a field. Then any ideal of the form I × J with I 6= (0) has only finitely many divisors. On the other hand, the prime ideal (0) × K has only one essential divisor.  For the statement of the following theorem, recall that a ring R is a general ZPI-ring if every proper ideal is a product of prime ideals, or equivalently R is a finite direct product of Dedekind domains and SPIRs. (See [16] or [27, Theorem 39.2].) Theorem 3.16. A ring is a Fletcher UIFR if and only if it is a general ZPI-ring. Q Proof. By Theorem 3.13, a finite direct product ni=1 Ri of rings is a Fletcher UIFR if and only if each Ri is a Fletcher UIFR. Because Fletcher UIFRs are completely decomposable (Theorem 3.6), we only need to prove the theorem for connected rings. (⇒): Let R be a connected Fletcher UIFR. Write the reduced atomic factorization of (0) in the form (0) = mk 1 Qm 1 · · · Qk , where the Qi ’s are distinct. Note that every non-faithful weakly nonfactorable ideal is an essential divisor of (0), hence is one of the Qi ’s. Therefore every non-faithful S ideal is divisible by some Qi , hence Z(R) = ki=1 Qi . We claim that each Qi is prime. Let xy ∈ Qi , and let (x) = I1 · · · Im and (y) = J1 · · · Jn be atomic factorizations. Then

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Q mj mk i −1 1 I1 · · · Im J1 · · · Jn Qm = (0) = Qm 1 · · · Qk . Reducing the left-hand side and i j6=i Qj applying uniqueness, we see that Qi ∈ {I1 , . . . , Im , J1 , . . . , Jn }, hence Qi ⊇ (x) or (y). With the claim established, it follows that every non-regular ideal is contained in one of the Qi ’s, hence is not faithful. Therefore the only non-regular weakly nonfactorable ideals are the Qi ’s. So each minimal prime is a Qi . Conversely, the Qi ’s are minimal primes, for otherwise there is a reduced atomic factorization of (0) not including all the Qi ’s. By considering reduced atomic factorizations, we see that each Qi is a multiplication ideal, i.e., divides each ideal it contains. Therefore the Qi ’s are locally principal [5, Theorem 1]. Next we claim that each invertible (weakly) nonfactorable ideal P is prime. Let xy ∈ P . There is a regular z ∈ P , and, because Z(R) is a finite union of prime ideals, there are s, t ∈ R with x+sz and y +tz regular [27, Exercise 7.8(c)]. Then (x+sz)(y +tz) ∈ P and (x+sz)(y +tz) = ((x+sz)(y +tz)P −1 )P . Replacing the factors in these factorizations with atomic factorizations, we get (necessarily reduced) atomic factorizations. By uniqueness, we conclude that P ⊇ (x + sz) or (y + tz), hence P ⊇ (x) or (y), as desired. Therefore every invertible proper ideal is a product of prime ideals. Now we claim that each localization of R at a maximal ideal M is either an SPIR or UFD. By [37] or [27, Theorem 46.11], it suffices to show that every principal proper ideal (x)M of RM is a product of prime ideals. Because the only non-regular weakly nonfactorable ideals of R are the Qi ’s, which are locally principal, we can write (x)M = (p1 · · · pn I)M (n ≥ 0), where I is a regular ideal and each (pi )M is prime. Since I is generated by its regular elements [27, Exercise 7.8(c)] and RM is quasilocal, we have (x)M = (p1 · · · pn y)M for some regular y ∈ I. Then (y) = R or a product of prime ideals, establishing the claim. Because R locally has a unique minimal prime, the Qi ’s are comaximal, hence Nil(R) = Q1 by connectedness. If some localization of R at a maximal ideal is an SPIR, then Q1 is the only prime ideal and R is an SPIR. So let us assume that R is locally a UFD. Then Q1 = (0) holds locally, hence globally. So R is a domain. Then every nonzero finitely generated ideal is half cancellation (Theorem 3.1(2)), hence has a unique atomic factorization. It easily follows that the monoid of finitely generated ideals of R is cancellative, so R is Pr¨ ufer [27, Theorem 24.3]. Since it is locally a UFD, it is almost Dedekind. Because R is J-Noetherian (Theorem 3.6), it is Dedekind. (⇐): Clear.  We have only been able to make partial progress on characterizing U-UIFRs. Theorem 3.17. (1) A non-connected ring is a U-UIFR if and only if it is a general ZPI-ring. (2) A pr´esimplifiable ring is a (U-)UIFR if and only if it is a Dedekind domain, an SPIR, or a quasilocal ring (R, M ) with M 2 = (0). Proof. Part (1) follows by combining Theorems 3.13 and 3.16. The UIFR case of (2) is [26, Theorems 7-8], so we only need to show that a pr´esimplifiable U-UIFR R is a UIFR. Each nonzero finitely generated proper ideal is half cancellation (Theorem 3.1(2)), hence has a unique atomic factorization. It easily follows that R satisfies the restricted finite cancellation law (RFCL), i.e., IA = JA 6= 0 with I, J, A finitely generated ideals ⇒ I = J. Therefore Nil(R) is prime [11, Theorem 6.43]. By Theorem 3.16, we may assume R is not a domain. Pick 0 6= x ∈ Nil(R). By pr´esimplifiability, we have L(x) = LU (x) ≤ LU ((x)) < ∞, hence x has an atomic factorization. Because Nil(R) is prime, some factor y is nilpotent, say y n = 0 6= y n−1 . We claim that any product x1 · · · xn of n nonunits is zero. We proceed by induction on the number k of factors not in (y). The case k = 0 is clear, so let us assume k ≥ 1, say x1 ∈ / (y). Induction yields x2 · · · xn y = 0. Note that ann(x2 · · · xn ) * (x1 ),

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since otherwise (y) ( (x1 ), contradicting irreducibility. (Here we use pr´esimplifiability.) But ann(x2 · · · xn ) is divided [11, Corollary 4.17], so (x1 ) ⊆ ann(x2 · · · xn ), as desired. Therefore (R, Nil(R)) is quasilocal with Nil(R)n = (0). Assume R is not an SPIR. Then (y) 6= (z) for some atom z. The preceding argument shows that z n = 0 6= z n−1 . Since (y) and (z) are nonfactorable (Theorem 3.10), uniqueness gives (z n−1 ) * (y), hence (z n−1 , y)(z) = (yz) = (0) by RFCL. Applying RFCL again gives (y, z)(z) = (z)2 = (0), hence n = 2.  Question 3.18. Are there any U-UIFRs besides general ZPI-rings and quasilocal rings (R, M ) with M 2 = (0)? We can classify the arithmetical rings satisfying various ideal U-factorization properties. Lemma 3.19 ([10]). Weakly nonfactorable ideals in arithmetical rings are prime. Proof. Let I be a weakly nonfactorable ideal of an arithmetical ring. We need to show that (I : (x)) = I for each x ∈ / I. The equation I = (I : (x))(I + (x)) holds locally, hence globally. Since I 6= I + (x), we have I = (I : (x)) by weak nonfactorability.  Theorem 3.20. (1) An arithmetical ring is weakly factorable (resp., a U-FIDR, a Fletcher FIDR, a UHIFR, a Fletcher HIFR, a U-UIFR, a Fletcher UIFR) if and only if it is a general ZPI-ring [10]. (2) An arithmetical ring is factorable if and only if it is a Dedekind domain or a finite direct product of non-reduced SPIRs [10]. (3) An arithmetical ring is a U-FIFR if and only if it is a Dedekind domain, a finite direct product of SPIRs, or a direct product of a Dedekind domain and a field. (4) An arithmetical ring is a Fletcher FIFR if and only if it is a Dedekind domain or a finite direct product of SPIRs. Proof. For part (1), combine Theorem 3.16 and Lemma 3.19. For parts (2), (3), and (4), combine (1) with Theorems 3.13, 3.15, and 3.14, respectively.  We have only been able to make partial progress on classifying the arithmetical rings that are U-BIFRs or Fletcher BIFRs. Theorem 3.21. Let R be a finite direct product of pr´esimplifiable arithmetical rings. Then R is a U- (resp., Fletcher) BIFR if and only if it is a general ZPI-ring. Proof. (⇒): By Theorem 3.13, we may assume R is a pr´esimplifiable arithmetical U-BIFR. Each nonzero finitely generated ideal I is half cancellation, hence L(I) = LU (I) < ∞ and I has an atomic factorization. By Lemma 3.19, every finitely generated ideal is a product of prime ideals, hence R is a general ZPI-ring by [34, Theorem 2.3]. (⇐): Theorem 3.20(1).  Question 3.22. Are there any arithmetical U-BIFRs besides general ZPI-rings? (If so, then Question 3.7 has a negative answer.) We next consider the interaction between U-factorization of ideals and localization. Lemma 3.23. Let I be a proper ideal of a ring R and S ⊆ R \ ZR (R/I) be multiplicative. Then LU (I) ≤ LU (IS ). Proof. It suffices to show that, if I = I1 · · · Ik is a reduced factorization, then so is IS = (I1 )S · · · (Ik )S . Each Ii ⊆ ZR (R/I), hence (Ii )S 6= RS . Suppose that the factorization IS = (I1 )S · · · (Ik )S is not reduced, say IS = (I2 )S · · · (Ik )S . Then for each x ∈ I2 · · · Ik there is an s ∈ S ⊆ R \ Z(R/I) with sx ∈ I, hence x ∈ I. So I = I2 · · · Ik , a contradiction. 

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S Lemma 3.24. Let I be a proper ideal P of a ring R with P ZR (R/I) ⊆ Qni=1 Pi for some {Pi }ni=1 ⊆ Spec(R). Then LU (I) ≤ min { ni=1 LU (IPi ), sup { ni=1 mi | I ⊆ ni=1 Pimi }}. Proof. Let I = I1 · · · Ik be a reduced factorization. Q For the second inequality, it suffices to show that each Ii is contained in some Pj . Since j6=i Ij 6= I, we have Ii ⊆ Z(R/I) ⊆ Sn Pj and the result follows. Lemma 3.23 shows that LU (I) ≤ LU (IS ), where S := j=1S Pn R \ ni=1 Pi . By passing to RS , we reduce to Q proving LU (I) ≤ i=1 LU (IPi ) in the case where Max(R) ⊆ {Pi }ni=1 . Then each IPi = j∈Λi (Ij )Pi for some Λi ⊆ {1, . . . , k} with Q |Λi | ≤ LU (IPi ). The equation I = j∈Sn Λi Ij holds locally, hence globally. Therefore i=1 S P k = | ni=1 Λi | ≤ ni=1 LU (IPi ).  Theorem 3.25. Let R be a ring, let I be a proper S ideal contained in only finitely many n maximal ideals M , . . . , M , and define S := R \ 1 i=1 Mi . Then LU (I) = LU (IS ) ≤ Pn Pnn Qn mi min{ i=1 LU (IMi ), sup { i=1 mi | I ⊆ i=1 Mi }}. Proof. For x ∈ / I we have (x)Mi * IMi for some i, hence (I : (x)) ⊆ Mi . So S ⊆ R \ ZR (R/I). In view of Lemmas 3.23 and 3.24, we only need to show that LU (I) ≥ LU (IS ). Let IS = (I1 )S · · · (Ik )S be a reduced factorization. We may assume each Ij ⊇ I. For each Mi , we have IMi = (IS )(Mi )S = ((I1 · · · Ik )S )(Mi )S = (I1 · · · Ik )Mi , and for each N ∈ Max(R) \ {Mi }ni=1 we have IN = RN = (I1 · · · Ik )N . Therefore I = I1 · · · Ik , and this factorization is certainly reduced, showing that LU (I) ≥ LU (IS ).  Theorem 3.26. A one-dimensional domain is (weakly) factorable (resp., a U-BIFD, a UFIDD, a U-HIFD, a U-UIFD) ⇔ it has finite character and locally satisfies this property. Proof. Each of the above properties implies J-Noetherian (Theorem 3.6), so we only need to consider a one-dimensional finite character domain D. • ((Weakly) factorable case.) (⇒): Assume D is weakly factorable and let M be a maximal ideal. Each nonzero proper ideal of DM can be written in the form IM with I M -primary. Let I = I1 · · · In be an atomic factorization. Each Ii is M -primary, hence (Ii )M is weakly nonfactorable (Theorem 3.25), hence IM = (I1 )M · · · (In )M is an atomic factorization. Since half cancellation is a local property (Theorem 3.1(1)), each (Ii )M is nonfactorable if D is factorable. (⇐): Assume D is locally weakly factorable. Let I be√a nonzero proper ideal and M1 , . . . , Mn be the maximal ideals containing I. Since I = M1 · · · Mn is a complete comaximal factorization, it follows from [35, Lemma 5.5] that the√complete comaximal factorization of I has the form I = Q1 · ·√ · Qn with each Qi + I = Mi . Because any prime ideal containing Qi also contains I, it follows that Qi is Mi -primary. Each atomic factorization of (Qi )Mi can be written in the form (Qi )Mi = (Ji,1 )Mi · · · (Ji,ki )Mi with each Ji,j Mi -primary. Each Q Qi Ji,j is weakly nonfactorable (Theorem 3.25), and the equation I = ni=1 kj=1 Ji,j holds locally, hence globally. Since half cancellation is a local property (Theorem 3.1(1)), each Ji,j is nonfactorable if D is locally factorable. • (U-BIFD case.) (⇒): Assume D is a U-BIFD. Let M be a maximal ideal and I be an M -primary ideal. Theorem 3.25 gives LU (IM ) = LU (I) < ∞. (⇐): Assume D is locally a U-BIFD. Let I be a nonzero proper ideal M1 , . . . , Mn be the maximal Pand n ideals containing I. Theorem 3.25 gives LU (I) ≤ i=1 LU (IMi ) < ∞. • (U-FIDD case.) (⇒): Assume D is a U-FIDD. Then D is locally weakly factorable. Let M be a maximal ideal and I be an M -primary ideal. Examining the proof of

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the “⇐” direction of the “weakly factorable” case, we see that each reduced atomic factorization of IM has the form IM = (I1 )M · · · (In )M , where I = I1 · · · In is a (necessarily reduced) atomic factorization. Therefore DM is a U-FIDD. (⇐): Assume D is locally a U-FIDD. Then D is weakly factorable. Let I, Mi , and Qi be as in the proof of the “⇐” direction of the “weakly factorable” case. Since weakly nonfactorable ideals are pseudo-irreducible, each atomic factorization of I is (up Q Qi Qi to order) of the form I = ni=1 kj=1 Ji,j with each kj=1 Ji,j = Qi [33, Corollary 4.9]. So we may assume I = Q1 . If I = I1 · · · Ik is a reduced atomic factorization, then IM1 = (I1 )M1 · · · (Ik )M1 is an atomic factorization by Theorem 3.25, and it is reduced by the proof of “⇒”. Because the Ii ’s are M1 -primary, different reduced atomic factorizations of I will lead to different reduced atomic factorizations of IM1 . Therefore I has only finitely many reduced atomic factorizations. • (U-HIFD and U-UIFD cases.) Adjust the proof of the U-FIDD case.  We now examine ideal factorization in idealizations. Recall that the idealization of a nonzero module A over a ring R is the ring R n A with underlying set R × A and operations (x, a) + (y, b) := (x + y, a + b) and (x, a)(y, b) := (xy, xb + ya). We will generally assume the reader has a basic familiarity with the technique of idealization. A fairly comprehensive reference on this topic was written by Anderson and Winders [14]. Our results work out most cleanly for idealizations over domains, but some of the theory can be generalized further. In order to state the next two theorems, we need to extend some of our factorization terminology from ideals to submodules. The set S(A) of R-submodules of A forms a module over the semiring I(R). We say C ∈ S(A) is a divisor of B ∈ S(A) if B = IC for some I ∈ I(R). We define the factorization length of B ∈ S(A) to be L(B) := sup {n ∈ N0 | B = I1 · · · In C, Ii ∈ I(R)# 0 , C ∈ S(A)}, and its essential factorization length is LU (B) := sup {n ∈ N0 | B = dI1 · · · In eC, Ii ∈ I(R)# 0 , C ∈ S(A)}. Here we write B = dI1 · · · In eC to indicate that the expression B = I1 · · · In C is irredundant, i.e., B 6= I1 · · · Ii−1 Ii+1 · · · In C for i = 1, . . . , n. Note that these definitions are consistent with our earlier definitions in the special case of ideal factorization. Lemma 3.27. Let R be a ring, A be a nonzero R-module, and define π1 : R n A → R : (x, a) 7→ x. (1) For J ∈ I(R n A) with π1 [J] 6= (0), we have L(J) ≤ L(π1 [J]). (2) For (0) 6= I ∈ I(R), we have L(I) = L(I n IA) and LU (I) ≤ LU (I n IA). (3) For 0 6= B ∈ S(A), we have L((0) n B) ≥ L(B) + 1 and LU ((0) n B) ≥ LU (B) + 1, with equality in both cases if R is a domain. Proof. (1) If J = J1 · · · Jn is a factorization, then each π1 [Ji ] 6= R and π1 [J] = π1 [J1 ] · · · π1 [Jn ]. (2) If I = I1 · · · In is a (reduced) factorization, then so is I nIA = (I1 nI1 A) · · · (In nIn A). Thus L(I) ≤ L(I n IA) and LU (I) ≤ LU (I n IA). We have L(I) ≥ L(I n IA) by (1). (3) We prove the “LU ” case. The “L” case can Q be shown with minor adjustments. If B = dI1 · · · In eC, then (0)nB = d((0)nC) ni=1 (Ii nIi A)e. Therefore LU ((0)nB) ≥ LU (B) + 1. Now assume R is a domain. To show that LU ((0) n B) ≤ LU (B) + 1, let (0) n B = J1 · · · Jn be a reduced factorization. Then π1 [J1 ] · · · π1 [Jn ] = (0), so some π1 [Ji ] = (0). Say J1 = (0) n C. Then B = π1 [J2 ] · · · π1 [Jn ]C, and it will suffice to show that this expression is irredundant. If not, say B = π1 [J3 ] · · · π1 [Jn ]C, then (0) n B = ((0) n C)J3 · · · Jn , a contradiction. 

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Theorem 3.28. Let D be a domain and A be a nonzero D-module. (1) If D is a BIFD, then D n A is (weakly) factorable if and only if every submodule of A has a divisor with (essential) factorization length zero [10]. (2) If D satisfies HCL, then D n A is a U-BIFR if and only if (i) D is a BIFD and (ii) LU (B) < ∞ for all 0 6= B ∈ S(A). (3) If D satisfies HCL, then D n A is a U-FIFR if and only if (i) D is an FIFD, (ii) each 0 6= B ∈ S(A) has only finitely many expressions of the form B = dI1 · · · In eC, and (iii) there is no infinite set of ideals of D n A with the same nonzero image under π1 : D n A → D : (x, a) 7→ x. (4) The ring D n A is a (U-)UIFR if and only if D is a field. Proof. (1) Assume D is a BIFD. (⇒): Assume D n A is (weakly) factorable and let 0 6= B ∈ S(A). Let (0) n B = J1 · · · Jn be a factorization with (weakly) nonfactorable factors. Because (0) n A is prime, some Ji has the form (0) n C, where C necessarily divides B. By Lemma 3.27(3), we see that C has (essential) factorization length zero. (⇐): Assume every submodule of A has a divisor with (essential) factorization length zero. For J ∈ I(D n A) with π1 [J] 6= (0) we have L(J) ≤ L(π1 [J]) < ∞, hence J is a product of nonfactorable ideals. Now consider an ideal of the form (0) n B. Write B = IC with C having (essential) factorization length zero. Then (0) n B = (I n A)((0) n C), where I n A is a product of nonfactorable ideals by the above argument, and (0) n C is (weakly) nonfactorable by Lemma 3.27(3). (2) Assume D satisfies HCL. (⇒): Assume D n A is a U-BIFR. Pick (0) 6= I ∈ I(D) and 0 6= B ∈ S(A). Lemma 3.27 gives L(I) = LU (I) ≤ LU (I n IA) < ∞ and LU (B) ≤ LU ((0) n B) < ∞. (⇐): Assume (i) and (ii) hold. For 0 6= B ∈ S(A), Lemma 3.27(3) gives LU ((0) n B) = LU (B) + 1 < ∞. So pick J ∈ I(D n A) with π1 [J] 6= (0). Lemma 3.27(1) gives L(J) ≤ L(π1 [J]) < ∞. (3) Assume D satisfies HCL. (⇒): Assume D n A is a U-FIFR. If I = I1 · · · In is a factorization of a nonzero ideal of D, then I n IA = (I1 n I1 A) · · · (In n In A) is a reduced factorization. If 0 6= B ∈ S(A) and B = dI1 · · · In eC, then (0) n B = d(I1 n A) · · · (In n A)((0) n C)e. The previous two sentences show that (i) and (ii) hold. Now consider any {Jλ }λ∈Λ ⊆ I(D n A) with each π1 [Jλ ] = I 6= (0). If IA = 0, then each Jλ2 = I 2 n 0, hence each Jλ is an essential divisor of I 2 n 0 and {Jλ }λ∈Λ is finite. If 0 6= IA ( A, then (0) n IA = dJλ ((0) n A)e for each λ, hence {Jλ }λ∈Λ is finite. So we may assume IA = A. We will show that P each Jλ = I n A by demonstrating that (0, a) ∈ Jλ for each a ∈ A. Write aP = ni=1 xi ai with xi ∈ I and ai ∈ A. Pick bi ∈ A with (xi , bi ) ∈ Jλ . Then (0, a) = ni=1 (xi , bi )(0, ai ) ∈ Jλ . (⇐): Assume (i)-(iii) hold. We need to show that each J ∈ I(D n A)# has only finitely many essential divisors. If π1 [J] 6= (0), then each divisor of J is one of the finitely many ideals whose image under π1 divides π1 [J]. So let us assume J = (0) n B for some 0 6= B ∈ S(A). Then each essential divisor of J is either an ideal of the form (0) n C with C one of the finitely many submodules of A dividing B, or one of the finitely many ideals whose image under π1 appears in an expression B = dI1 · · · In eC. (4) (⇒): Assume D n A is a U-UIFR. We first claim that D is a UIFD. As in the proof of Theorem 3.17(2), it suffices to show that each nonzero finitely generated proper ideal I has a unique atomic factorization. Since I is half cancellation (Theorem 3.1(2)), we have L(I) = LU (I) ≤ LU (I nIA) < ∞ by Lemma 3.27(2). Therefore I has an atomic

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J.R. JUETT AND C.P. MOONEY

factorization. For uniqueness, let I1 · · · Im = J1 · · · Jn be atomic factorizations of I. Each factor is half cancellation, hence nonfactorable. So (I1 n I1 A) · · · (Im n Im A) = (J1 nJ1 A) · · · (Jn nJn A) are atomic factorizations of I nIA by Lemma 3.27(2). Since I n IA is half cancellation, these factorizations are reduced. Applying uniqueness, we obtain m = n and each Ii = Ji after a suitable reordering. Next we claim that IA = A for each (0) 6= I ∈ I(D). Because D is factorable, we may assume I is nonfactorable. If IA = 0, then (I n A)2 = (I n 0)2 are atomic factorizations of I 2 n 0 (Lemma 3.27(1)) without a common subfactorization, a contradiction. So (I n A)((0) n A) = (I n IA)((0) n A) are atomic factorizations of the nonzero ideal (0) n IA. These factorizations are either reduced or (0) n A = (0) n IA. Either way, we have A = IA. Now we show that Z(A) = (0). Suppose to the contrary that xa = 0 with 0 6= x ∈ D and 0 6= a ∈ A. By (1), we have Da = IB with LU (B) = 0. Pick 0 6= y ∈ I. Because A = xyA, we can write B = xyC for some C ∈ S(A). But LU (B) = 0, so C = B and B = xyB ⊆ xIB = Dxa = 0, a contradiction. Therefore D n A is pr´esimplifiable [14, Theorem 5.1(1)]. Theorem 3.17(2) then implies (D, M ) is quasilocal and either D n A is an SPIR or (M n A)2 = 0. The latter case implies M 2 = (0), hence D is a field. It is a field in the former case by [14, Theorem 4.10]. (⇐): If D is a field, then (D n A, (0) n A) is quasilocal with ((0) n A)2 = 0.  We are finally ready to give the promised examples associated with Figure 1 and its “Fletcher” analog. We also explore the relationship between the “Fletcher” properties, the “U-” properties, and the properties without a “Fletcher” or “U-”. Example 3.29. No nontrivial implications can be added to Figure 1 or its “Fletcher” analog, with the possible exception that U-BIFR (or Fletcher BIFR) might imply weakly factorable. Furthermore, each “U-” property is strictly weaker than the corresponding “Fletcher” property and the corresponding property without a “U-” or “Fletcher”. Finally, there are no implications between a given “Fletcher” property and the corresponding property without the “Fletcher”. (1) (Fletcher UIFR ; factorable, wndf-ring, or U-FIFR): By Theorems 3.16, 3.13(2), and 3.15, the ring Z × Z is a Fletcher UIFR, but not factorable or a U-FIFR. It is not a wndf-ring since (p) × Z is a nonfactorable divisor of (0) × Z for each prime p 6= 0. (2) (UIFR ; Fletcher wndf-ring): Let (R, M ) be a non-Noetherian quasilocal ring with M 2 = (0), e.g., R := Q[X1 , X2 , . . .]/(X1 , X2 , . . .)2 . Each of its infinitely many nonzero proper ideals is a nonfactorable essential divisor of (0), so R is a (U-)UIFR but not a Fletcher wndf-ring. (3) (FIFD ; U-HIFR): Let K be a finite field. Then D := K + X 2 K[[X]] is a CKdomain [12, Theorem 4.5(1)], i.e., a (necessarily Noetherian) domain with I(D) a finitely generated monoid [6, Theorem 1]. Since D satisfies HCL (Theorem 3.1(2)) and is a U-FIDD (Theorem 3.5), it is in fact a (Fletcher) FIFD. Because D is local and X 2 and X 3 are irreducible, the ideals (X 2 ) and (X 3 ) are nonfactorable (Theorem 3.10). Since (X 2 )3 = (X 3 )2 , we see that D is not a U-HIFR. (4) (Fletcher FIFR ; factorable.) The ring Q × Q is a Fletcher FIFR (Theorem 3.14) but not factorable (Theorem 3.13(2)). (5) (wndf-domain ; weakly factorable or U-BIFR): A one-dimensional non-discrete valuation domain is a (Fletcher) wndf-domain (Lemma 3.19), but neither weakly factorable (Theorem 3.20(1)) nor a U-BIFR (Theorem 3.21).

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(6) (HIFD ; U-wndf-ring): Let K ( and D := K +XL[[X]]. Because T L be infinite fields n (D, XL[[X]]) is quasilocal and ∞ (XL[[X]]) = (0), it follows that D is a BIFD. n=1 But D is not an FIFD [31, Theorem 4.3], hence is not a (U-)wndf-ring (c.f. [13, Proposition 6.6]). To show that D is an HIFD, let I1 · · · Im = J1 · · · Jn be atomic factorizations of a nonzero ideal. Note that IL[[X]] = XL[[X]] for each nonzero nonfactorable ideal I of D. (For otherwise IL[[X]] ⊆ X 2 L[[X]] and XD properly divides I, a contradiction.) Therefore X m L[[X]] = X n L[[X]] and m = n. (7) (HIFR ; Fletcher HIFR): Let (R, (p)) be an SPIR with p2 = 0 6= p, e.g., R := Z/4Z. Then R1 := R n R is local with maximal ideal M := (p) n R. Because M 3 = 0, it follows that L(I) ≤ 2 for each nonzero ideal I. Therefore R1 is a (U-)HIFR. But J := (0) n R is nonfactorable since it is not contained in M 2 = (0) n (p), so M 3 = J 2 are reduced atomic factorizations and R1 is not a Fletcher HIFR. (8) (BIFR ; P Fletcher BIFR):PLet K be a field, D := K[X1 , X2 , . . .], M := (X1 , X2 , . . .), k+1 and I := i