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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.

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

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 ⁄ 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.

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.

Classical ideal theory is based on the work of Krull, Noether, Prufer and other researchers – these are collated in [68]).

The class of Noetherian rings has an extremely important significance to geometric applications.

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).

Let D be an integral domain with quotient field K, X be an indeterminate over D, Γ be a numerical semigroup with Γ ⊊ ℕ0, D[Γ] be the semigroup ring of Γ over D (and hence D ⊊ D[Γ] ⊊ D[X]), and D + X n K[X] = {a + X n g∣a ∈ D and g ∈ K[X]}. We show that there exists an order-preserving bijection between Spec(D[X]) and Spec(D[Γ]), which also preserves t-ideals. We also prove that D[Γ] is an APvMD (resp., AGCD-domain) if and only if D[X] is an APvMD (resp., AGCD-domain) and char(D) ≠ 0. We show that if n ≥ 2, then D is an APvMD (resp., AGCD-domain, AGGCD-domain, AP-domain, AB-domain) and char(D) ≠ 0 if and only if D + X n K[X] is an APvMD (resp., AGCD-domain, AGGCD-domain, AP-domain, AB-domain). Finally, we give some examples of APvMDs which are not AGCD-domains by using the constructions D[Γ] and D + X n K[X].

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.

For an extension A ⊆ B of commutative rings, we present a sufficient condi-tion for the ring [[A S, ≤ ]] of generalized power series to be weakly normal (resp., stronglyt-closed) in [[B S, ≤ ]], where (S, ≤) be a torsion-free cancellative strictly ordered monoid.As a corollary, it can be applied to the ring of power series in infinitely many indetermi-nates as well as in finite indeterminates. 1. Introduction and preliminariesLet A ⊆ B be an extension of commutative rings with (the same) identity.Consider the following conditions:(a) B is integral over A.(b) Spec(B) → Spec(A) is a bijection.(c) The residue field extensions are isomorphisms. i.e., for each Q ∈ Spec(B) theextension A P /PA P ,→ B Q /QB Q is an isomorphism, where P = Q∩A.(c 0 ) The residue field extensions are purely inseparable.We first recall some special extensions satisfying two or three conditions aboveincluding the condition (a).• R. G. Swan called the extension A ⊆ B subintegral if (a), (b) and (c) aresatisfied.

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 introduce the concept of fuzzy star-operations on an integral domain and show that the set of all fuzzy star-operations on the integral domain forms a complete lattice. We also characterize Prufer domains, psuedo-Dedekind domains, (generalized-) greatest common divisor domains, and other integral domains in terms of the invertibility of certain fractionary fuzzy ideals.

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.

In this paper, we determine necessary and sufficient conditions for the group ring D[G] to be a BFD (resp., an FFD, an SFFD). Also we give necessary and sufficient conditions for the monoid domain D[S] to be a BFD (resp., an FFD, an SFFD). In addition, we characterize when themonoid domain D[S] is a UFD in terms of 2-factoriality.

In this paper, we show that any nonzero divisor class of a Krull domain D[S] contains a height-one prime ideal of D[S], where either D is a UFD or S is a group.

In this paper we propose a key-exchange system and a public-key encryption scheme based on the class semigroups of imaginary quadratic non-maximal orders, the former is analogous to the Diffie-Hellman's key-exchange system and the latter is similar to the ElGamal's encryption scheme, whose security is based on the difficulty of the discrete logarithm problem of that class semigroup.

We show how to use ideal arithmetic in the divisor class group of an affine normal subring of K[X, Y] generated by monomials, where K is a field, to design new public-key cryptosystems, whose security is based on the difficulty of the discrete logarithm problem in the divisor class group of that integral domain.