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Let $R$ be a commutative ring and $M$ be an $R$-module. A submodule $N$ of $M$ is called a d-submodule $($resp., an fd-submodule$)$ if $\ann_R(m)\subseteq \ann_R(m')$ $($resp., $\ann_R(F)\subseteq \ann_R(m'))$ for some $m\in N$ $($resp., finite subset $F\subseteq N)$ and $m'\in M$ implies that $m'\in N.$ Many examples, characterizations, and properties of these submodules are given. Moreover, we use them to characterize modules satisfying Property T, reduced modules, and von Neumann regular modules.

Let $R$ be a commutative ring with nonzero identity. An $R$-module $M$ is called $\phi$-P-flat if $x \in \Ann(s)M$ for every non-nilpotent element $s \in R$ and $x\in M$ such that $sx=0$. In this paper, we introduce and study the class of $\phi$-PF-rings, i.e., rings in which all ideals are $\phi$-P-flat. Among other results, the transfer of the $\phi$-PF-ring to the amalgamation is investigated. Several examples which delineate the concepts and results are provided.

An integral domain [Formula: see text] is said to be locally G-Dedekind if the Gorenstein global dimension of [Formula: see text] is at most one for each maximal ideal [Formula: see text]. In this paper, we show that an integral domain [Formula: see text] is not necessarily a G-Prüfer even if [Formula: see text] is a locally G-Dedekind domain, which gives a negative answer to the question raised by the first author. It follows that the localization of the G-Prüfer domain differs from the classical case of the Prüfer domain. We also study coherent locally G-Dedekind domains, called almost G-Dedekind domains. The almost G-Dedekind domains need not be integrally closed and fill the gap between the G-Dedekind domains and the G-Prüfer domains. Various examples are provided to illustrate the new concept.

Let [Formula: see text] and [Formula: see text] be two commutative rings graded by an arbitrary commutative monoid [Formula: see text], let [Formula: see text] be a homogeneous ideal of [Formula: see text] and [Formula: see text] a graded ring homomorphism. In this paper, we introduce a new graded ring construction called “the graded amalgamation of [Formula: see text] with [Formula: see text] along [Formula: see text] with respect to [Formula: see text]” (denoted by [Formula: see text]). Then we investigate, through the properties of the homogeneous components that build the body of this “graded” study, the transfer of the property of graded-coherence to the graded amalgamation [Formula: see text].

In this article, we introduce two new classes of modules over a ϕ \phi -ring that generalize the classes of coherent modules and Noetherian modules. We next study the possible transfer of the properties of being nonnil-Noetherian rings, ϕ \phi -coherent rings, and nonnil-coherent rings in the amalgamated algebra along an ideal.

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 $$f : A \longrightarrow B$$ be a ring homomorphism and let J be an ideal of B. In this article, we study the possible transfer of the properties of being a $$\phi $$ -ring, a $$\phi $$ -chained ring, and a $$\phi $$ -pseudo-valuation ring to the amalgamated algebra $$A\bowtie ^{f}J$$ . Our aim is to provide examples of new classes of commutative rings satisfying the aforementioned properties.

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...

We first characterize unique factorization domains of power series. Then we study S-invertible ideals of domains of formal power series. The concept of the S-Picard group of an integral domain is i...

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, ∗ 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.

Let D be an integral domain, V (D) (resp., t-V (D)) be the set of all valuation (resp., t-valuation) ideals of D, and w-P(D) be the set of primary w-ideals of D. Let D[X] be the polynomial ring over D, c(f) be the ideal of D generated by the coefficients of f ∈ D[X], and Nv = {f ∈ D[X] | c(f)v = D}. In this paper, we study integral domains D in which w-P(D) ⊆ t-V (D), t-V (D) ⊆ w-P(D), or t-V (D) = w-P(D). We also study the relationship between t-V (D) and \(V\left( {D{{\left[ X \right]}_{{N_v}}}} \right)\), and characterize when t-V (A + XB[X]) ⊆ w-P(A + XB[X]) holds for a proper extension A ⊂ B of integral domains.

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.

It is well known that an integral domain D is a UFD if andonly if every nonzero prime ideal of D contains a nonzero principal prime.This is the so-called Kaplansky’s theorem. In this paper, we give this typeof characterizations of a graded PvMD (resp., G-GCD domain, GCDdomain, B´ezout domain, valuation domain, Krull domain, π-domain). 0. IntroductionThis is a continuation of our works on Kaplansky-type theorems [13, 20].It is well known that an integral domain D is a UFD if and only if everynonzero prime ideal of D contains a nonzero principal prime [19, Theorem 5].This is the so-called Kaplansky’s theorem. A generalization of this type oftheorems was first studied by Anderson and Zafrullah in [5], where they gaveseveral characterizations of this type for GCD domains, valuation domains, andPru¨fer domains. Also, characterizations for PvMDs and Krull domains weregiven in [15] and [6], respectively.Later, in [20], the second-named author gave a Kaplansky-type charac-terization of G-GCD domains and PvMDs and gave an ideal-wise version ofKaplansky-type theorems. This ideal-wise version is then used to give char-acterizations of UFDs, π-domains, and Krull domains. Let D be an integraldomain with quotient field K, X be an indeterminate over D, and D[X] bethe polynomial ring over D. A nonzero prime ideal Q of D[X] is called anupper to zero in D[X] if Q ∩ D = (0). Clearly, Q is an upper to zero in D[X]if and only if Q = fK[X] ∩ D[X] for some nonzero polynomial f ∈ D[X].For f ∈ D[X], let c(f) be the ideal of D generated by the coefficients of f.In [13], the first two authors of this paper studied an integral domain D suchthat every upper to zero in D[X] contains a prime (resp., primary) element

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.

In this paper, we characterize w-Noetherian modules in terms of polynomial modules and w-Nagata modules. Then it is shown that for a finite type w-module M , every w-epimorphism of M onto itself is an isomorphism. We also define and study the concepts of w-Artinian modules and w-simple modules. By using these concepts, it is shown that for a w-Artinian module M , every w-monomorphism of M onto itself is an isomorphism and that for a w-simple module M , EndRM is a division ring.

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.

Let D be an integral domain, S be a (not necessarily saturated) multiplicative subset of D, w be the so-called w-operation on D, and M be a unitary D-module. As generalizations of strong Mori domains (respectively, UFDs) and strong Mori modules, we define D to be an S-strong Mori domain (respectively, S-factorial domain) if for each nonzero ideal I of D, there exist an s ∈ S and a w-finite type (respectively, principal) ideal J of D such that s I ⊆ J ⊆ I w ; and M to be an S-strong Mori module if M is a w-module and for each nonzero submodule N of M, there exist an s ∈ S and a w-finite type submodule F of N such that s N ⊆ F ⊆ N w . This paper presents some properties of S-strong Mori domains, S-factorial domains and S-strong Mori modules.

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.

Let R be an integral domain with quotient field F. It is shown that R is a strongly discrete Prufer v-multiplication domain if and only if there exists a bijection between the set of the prime w-ideals and the set of isomorphism classes of GV-torsionfree indecomposable injective R-modules and every indecomposable injective R-module, viewed as a module over its endomorphism ring, is uniserial. It is also shown that the w-closure of any GV-torsionfree homomorphic image of F is injective if and only if R is a Prufer v-multiplication domain satisfying an almost maximality-type property.

Let D be an integral domain, D[X] be the polynomial ring over D, and w be the so-called w-operation on D. We define D to be a w-LPI domain if every nonzero w-locally principal ideal of D is w-invertible. This article presents some properties of w-LPI domains. It is shown that D is a w-LPI domain if and only if D[X] is a w-LPI domain, if and only if D[X] N v is an LPI domain, where N v = {f ∈ D[X] | c(f)−1 = D}. As a corollary, it is also shown that D is a w-LPI domain if and only if each w-locally finitely generated and w-locally free submodule of D n of rank n over D is of w-finite type. We show that if is a finite character intersection of t-linked overrings D α and if each D α is a w-LPI domain, then D is a w-LPI domain.

In this article, we characterize t-Prufer modules in the class of faithful multi- plication modules. As a corollary, we also characterize Krull modules. Several properties of a t-invertible submodule of a faithful multiplication module are given.

In this article, we generalize the concepts of several classes of domains (which are related to Krull domains) to torsion-free modules, and show that for a faithful multiplication module M over an integral domain R, M is a Krull module if and only if R is a Krull domain. Then we characterize Krull, Dedekind, and factorial modules.

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

In this paper, we introduce and study the concept of \strongly t-linked ex- tensions", which is a stronger version of t-linked extensions of integral domains. We show that for an extension of Prufer v-multiplication domains, this concept is equivalent to that of \w-faithfully at".

Let RT be an extension of integral domains, X be an indeterminate over T, and R(X) and T(X) be polynomial rings. Then R � T is said to be LCM-stable if (aR\ bR)T = aT\ bT for all 0 6 a,b 2 R. Let wA be the so-called w-operation on an integral domain A. In this paper, we introduce the notions of w(e)- and w-LCM stable extensions: (i) RT is w(e)-LCM-stable if ((aR\ bR)T)wT = aT \ bT for all 0 6 a,b 2 R and (ii) RT is w-LCM-stable if ((aR \ bR)T)wR = (aT \ bT)wR for all 0 6 a,b 2 R. We prove that LCM-stable extensions are both w(e)-LCM- stable and w-LCM-stable. We also generalize some results on LCM-stable extensions. Among other things, we show that if R is a Krull domain (resp., PvMD), then RT is w(e)-LCM-stable (resp., w-LCM-stable) if and only if R(X) � T(X) is w(e)-LCM-stable (resp., w-LCM-stable).

It is shown that generalized GCD domains satisfy a certain property of injectivity and conversely this property characterizes generalized GCD domains.

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.