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For a property X of integral domains, D is said to be an MZ-X domain if DP has the property X for all associated prime ideals P of an integral domain D. Our first goal is to study some ring-theoret...

Let R be a commutative ring and w be the so-called w-operation on R. Then we introduce and study two concepts of w-FP-injective modules and w-IF rings. To do so, we use two main methods, of which o...

The purpose of this paper is to introduce two new classes of rings that is closely related to the classes of piecewise Noetherian domains and piecewise w-Noetherian domains. Let H={R|R is a commuta...

Let D be an integral domain, F+(D) (resp., f+(D)) be the set of nonzero (resp., nonzero finitely generated) ideals of D, R1 = f+(D) ∪ {(0)}, and R2 = F+(D) ∪ {(0)}. Then (Ri, ⊕, ⊗) for i = 1, 2 is a commutative semiring with identity under I ⊕ J = I + J and I ⊗ J = IJ for all I, J ∈ Ri. In this paper, among other things, we show that D is a Prüfer domain if and only if every ideal of R1 is a k-ideal if and only if R1 is Gaussian. We also show that D is a Dedekind domain if and only if R2 is a unique factorization semidomain if and only if R2 is a principal ideal semidomain. These results are proved in a more general setting of star operations on D.

Let D be an integral domain with quotient field K, [Formula: see text] be the integral closure of D in K, and D[w] be the w-integral closure of D in K; so [Formula: see text], and equality holds when D is Noetherian or dim(D) = 1. The Mori–Nagata theorem states that if D is Noetherian, then [Formula: see text] is a Krull domain; it has also been investigated when [Formula: see text] is a Dedekind domain. We study integral domains D such that D[w] is a Krull domain. We also provide an example of an integral domain D such that [Formula: see text], t-dim(D) = 1, [Formula: see text] is a Prüfer v-multiplication domain with t-dim([Formula: see text]) = 2, and D[w] is a UFD.

In this article, a new semistar operation, called the q-operation, on a commutative ring R is introduced in terms of the ring Q0(R) of finite fractions. It is defined as the map q:Fq(R)→Fq(R) by A↦...

An R-module M is called strongly FP-injective if ExtRi(P,M)=0 for any finitely presented R-module P and any i > 0. Denoted by SFI the class of all strongly FP-injective R-modules and by ⊥SFI the le...

Let R be a commutative ring with identity. An R-module E is said to be weak injective if ExtR1(N,E)=0 for any super finitely presented R-module N. In this article, for a domain R with quotient fiel...

Let D be an integral domain and w be the so-called w-operation on D. We define D to be a w-FF domain if every w-flat w-ideal of D is of w-finite type. This paper presents some properties of w-FF domains and related domains. Among other things, we study the w-FF property in the polynomial extension, the t-Nagata ring and the pullback construction.

Let D be an integral domain, ∗ a star-operation on D, and S a multiplicative subset of D. We define D to be an S-∗w-principal ideal domain if for each nonzero ideal I of D, there exist an element s ∈ S and a principal ideal (c) of D such that [Formula: see text]. In this paper, we study some properties of S-∗w-principal ideal domains. Among other things, we study the local property, the Nagata type theorem, and the Cohen type theorem for S-∗w-principal ideal domains.

In this paper, we survey David Anderson’s work on graded integral domains, with emphasis on Picard groups of graded integral domains, graded Krull domains, graded Prufer v-multiplication domains, graded Prufer domains, Nagata rings, and Kronecker function rings.

It is well known that the equivalence of injectivity and divisibility characterizes the Dedekind domains. In this note, we shed more light on this problem and show that a similar property characterizes Krull domains and π-domains.

In this article, we give a complete characterization of when the (composite) semigroup ring is a generalized Krull domain.

Let [Formula: see text] be a commutative ring with identity. In this paper, a Cohen-type theorem for [Formula: see text]-Artinian modules is given, i.e. a [Formula: see text]-cofinitely generated [Formula: see text]-module [Formula: see text] is [Formula: see text]-Artinian if and only if [Formula: see text] is [Formula: see text]-cofinitely generated for every prime [Formula: see text]-ideal [Formula: see text] of [Formula: see text]. As a by-product of the proof, we also obtain a detailed representation of elements of a [Formula: see text]-module and the [Formula: see text]-theoretic version of the Chinese remainder theorem for both modules and rings.

In this paper, the concept of almost projective covers is introduced. Using this concept, locally perfect rings are characterized. Also, the concepts of [Formula: see text]-almost projective modules, [Formula: see text]-almost projective covers, and [Formula: see text]-locally perfect rings are introduced. Then [Formula: see text]-locally perfect rings are characterized in terms of [Formula: see text]-almost projective modules and [Formula: see text]-almost projective covers, respectively.

Let R be a commutative ring, M be an R-module, and w be the so-called w-operation on R. Set S w = { f ∈ R [ X ] | c ( f ) w = R } , where c ( f ) denotes the content of f. Let R { X } = R [ X ] S w and M { X } = M [ X ] S w be the w-Nagata ring of R and the w-Nagata module of M respectively. Then we introduce and study two concepts of w-projective modules and w-invertible modules, which both generalize projective modules. To do so, we use two main methods of which one is to localize at maximal w-ideals of R and the other is to utilize w-Nagata modules over w-Nagata rings. In particular, it is shown that an R-module M is w-projective of finite type if and only if M { X } is finitely generated projective over R { X } ; M is w-invertible if and only if M { X } is invertible over R { X } . As applications, it is shown that R is semisimple if and only if every R-module is w-projective and that, in a Q 0 -PVMR, every finite type semi-regular module is w-projective.

We introduce the concept of w-zero-divisor (w-ZD) rings and study its related rings. In particular it is shown that an integral domain R is an SM domain if and only if R is a w-locally Noetherian w-ZD ring and that a commutative ring R is w-Noetherian if and only if the polynomial ring in one indeterminate R[X] is a w-ZD ring. Finally we characterize universally zero divisor rings in terms of w-ZD modules.

For a UMT-domain $D$, we characterize when the polynomial ring $D[X]$ is $t$-compactly packed and every prime $t$-ideal of $D[X]$ is radically perfect. As a corollary, for a quasi-Pr\"ufer domain $D$, we also characterize when every prime ideal of $D[X]$ is radically perfect. Finally we introduce the concepts of Serre's conditions in strong Mori domains and characterize Krull domains and almost factorial domains, respectively.

Let Rbe a commutative ring with identity. An R-module Mis said to be w-projective if Ext 1R (M,N) is GV-torsion for any torsion-freew-module N. In this paper, we define a ring R to be w-semi-hereditaryif every finite type ideal of R is w-projective. To characterize w-semi-hereditary rings, we introduce the concept of w-injective modules andstudy some basic properties of w-injective modules. Using these concepts,we show that Ris w-semi-hereditary if and only if the total quotient ringT(R) of R is a von Neumann regular ring and R m is a valuation domainfor any maximal w-ideal mof R. It is also shown that a connected ring Ris w-semi-hereditary if and only if Ris a Pru¨fer v-multiplication domain. 1. IntroductionThroughout, R denotes a commutative ring with identity 1 and E(M) de-notes the injective hull (or envelope) of an R-module M. And let us regardthat the v-, t- and w-operation are well-known star-operations on domains. Forunexplained terminologies and notations, we refer to [3, 14, 15].Pru¨fer v-multiplication domains (PVMD for short) have received a good dealof attention in much literature. A domain R is called a PVMD if every nonzerofinitely generated ideal I is t-invertible, that is, there is a fractional ideal B ofR such that (IB)

Let [Formula: see text] be a commutative ring. In this paper, a class of almost projective modules is introduced. An [Formula: see text]-module [Formula: see text] is said to be almost projective if [Formula: see text] for any [Formula: see text]-module [Formula: see text], where [Formula: see text] is a maximal ideal of [Formula: see text]. It is shown that an [Formula: see text]-module [Formula: see text] satisfying that [Formula: see text] is free over [Formula: see text] for any maximal ideal [Formula: see text] of [Formula: see text] is exactly almost projective. As applications, we characterize almost Dedekind domains in terms of almost projectivity and certain divisibility, respectively.

Let ⁄ be a star-operation on an integral domain R. We show that if R is a ⁄w-Noetherian domain, then quasi-⁄w-invertible prime ⁄w-ideals of R are minimal, and prime ideals of R minimal over a ⁄w-invertible ⁄w-ideal are minimal.

We give some characterizations of injective modules over w-Noetherian rings. It is also shown that each localization of a GV-torsion-free injective module over a w-Noetherian ring is injective. 1. IntroductionOne of bad properties of injective modules is that they do not preservelocalization in general. However, there are some positive results as follows.If a (commutative) ring R is Noetherian or hereditary, then localizations ofinjective R-modules are also injective ([10, Theorem 4.88] or [1, Proposition 2]).Recently Couchot investigated localizations of injective modules over valuationrings ([1]) and arithmetical rings ([2]). The purpose of this article is to provideanother positive result if R is a w-Noetherian ring.We first introduce some definitions and notation from [12, 16]. Throughoutwe let R be a commutative ring with identity. For an R-module M, the dualmodule Hom R (M,R) of M is denoted by M ♭ . Following [16, Definition 1.1],an ideal J of a commutative ring R is called a Glaz-Vasconcelos ideal or aGV-ideal, denoted by J ∈ GV(R), if J is finitely generated and the naturalhomomorphism φ : R → J

Let D be an integral domain, D be the integral closure of D, and 0 be a numerical semigroup with0 ( N0. Let t be the so-called t-operation on D. We will say that D is an AK-domain (resp., AUF-domain) if for each nonzero ideal.fa g/ of D, there exists a positive integer nD n.fa g/ such that.fa n g/t is t-invertible (resp., principal). In this paper, we study several properties of AK-domains and AUF-domains. Among other things, we show that if D D is a bounded root extension, then D is an AK-domain (resp., AUFdomain) if and only if D is a Krull domain (resp., Krull domain with torsion t-class group) and D is t-linked under D. We also prove that if D is a Krull domain (resp., UFD) with char. D/6D0, then the (numerical) semigroup ring DT0U is a nonintegrally closed AK-domain (resp., AUF-domain).

In 1977, Glaz and Vasconcelos [73] introduced the concept of semidivisorial modules to study some properties of flat modules.

Statements and research on module theory and homology theory closely related to the theory and methods of categories, and so the introduction of the concept of some categories is beneficial.

Algebraic K-theory extends many of the methodologies of linear algebra over a number field to over a ring R.

This chapter describes some theories of ring extensions, especially of integral extension. The concept of integral dependence, originally proposed by Noether, is a very useful tool for multiplicative ideal theory, but it appears isolated and it has failed to attract attention.

There are a lot of papers that deal with pullbacks. Milnor squares are actually very important pullbacks in applications. Milnor squares’ success in producing interesting examples has resulted in a good deal of attention in multiplicative ideal theory.

All the rings in this book are commutative. Sometimes we may also assume that an algebra may be non-commutative.

One of the famous results characterized by homological tricks is that a regular local ring is a UFD.

Relative homological algebra was conceived by Auslander and Bridger [14, 15] and was formed by Enochs, Jenda, and Torrecillas (for example, [54, 55, 57]).

The following statements for an infra-Krull domain R are shown to be equivalent: (1) R is a Krull domain; (2) for any essentially nite w-module M over R, the torsion submodule t(M) of M is a direct summand of M; (3) for any essentially nite w-module M over R, t(M)\ pM = pt(M), for all maximal w-ideal p of R; (4) R satises the w-radical formula; (5) the R-module R R satises the w-radical formula.

Let be a star-operation on an integral domain R, and let I + � (R) be the semigroup of -invertible integral -ideals of R. In this article, we introduce the concept of a -coatom, and we then characterize when I + � (R) is a free semigroup with a set of free generators consisting of -coatoms. In particular, we show that I + � (R) is a free semigroup if and only if R is a Krull domain and each v-invertible v-ideal is -invertible. As a corollary, we obtain some characterizations of -Dedekind domains.

In this paper, we study w-nullity and (co-)semi-divisoriality of Hom-modules and the semi-divisorial envelope of HomR(M;N) under suitable conditions on R;M; and N. We also investigate an injective cogenerator of a quotient category.

If R is an integral domain, we extend to any R-module the notion of semi-divisorial envelope, or w-envelope, defined by Wang and McCasland for torsion-free R-modules, and introduce and study the related notions of co-semi-divisoriality and w-nullity. These concepts are then used to give new module-theoretic characterizations of t-linkative domains, a class of domains widely considered in multiplicative ideal theory.

In this paper, we define the v -finiteness for a length function L v on the set of all v -ideals of an integral domain R and show that R is a Krull domain if and only if every proper integral v -ideal of R has v -finite length and L v ( ( A B ) v ) = L v ( A ) + L v ( B ) for every pair of proper integral v -ideals A and B in R . We also give Euclidean-like characterizations of factorial, Krull, and π -domains. Finally we define the notion of quasi- ∗ -invertibility and show that if every proper prime t -ideal of an integral domain R is quasi- t -invertible, then R is a Krull domain.

For an extension A⊆B of commutative rings, we present a sufficient condition for the ring [[AS,⩽]] of generalized power series to be t-closed in [[BS,⩽]], where (S,⩽) is a torsion-free cancellative ordered monoid. As a corollary, this result can be applied to the ring of power series in any number of indeterminates.

satisfies the ACC on cyclic submonoids. On the otherhand, there are at least three distinct power series rings in infinitely manyindeterminates in the literature. In 5 , Gilmer proved that each of thesethree types of power series rings in infinitely many indeterminates is also aKrull domain in case the coefficient ring is a Krull domain. In 10 ,Ribenboim investigated the following question: If a ring

In this paper, we determine some sufficient conditions for an A + XB[X] domain to be an HFD. As a consequencewe give new examples of HFDs of the type A + XB[X].

A torsion-free module M over a commutative integral domain R is said to be LCM-stable over R if (Ra ∩ Rb)M = Ma ∩ Mb for all a, b ∈ R. We show that if the module M is LCM-stable over a GCD-domain R, then the polynomial module M[X] is LCM-stable over R[X]; if R is a w-coherent locally GCD-domain, then LCM-stability and reflexivity are equivalent for w-finite type torsion-free R-modules. Finally, we introduce the concept of w-LCM-stability for modules over a domain. Then we characterize when the module M is w-LCM-stable over the domain in terms of localizations and t-Nagata modules, respectively. Also we characterize Prüfer v-multiplication domains and Krull domains in terms of w-LCM-stability.