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An integral domain R is said to be locally G-Dedekind if the Gorenstein global dimension of R is at most one for each maximal ideal . In this paper, we show that an integral domain R is not necessarily a G-Prüfer even if R 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. [ABSTRACT FROM AUTHOR]

We study some homological properties of Gorenstein FP ∞ -injective modules, and we prove (1) a ring R is not necessarily coherent if every Gorenstein FP ∞ -injective R -module is injective, and (2) a ring R is not necessarily coherent if every Gorenstein injective R -module is injective. In addition, we characterize w -Noetherian rings in terms of Gorenstein FP ∞ -injective modules, and we prove that a ring R is w -Noetherian if and only if every GV-torsion-free FP-injective R -module is Gorenstein FP ∞ -injective, if and only if any direct sum of GV-torsion-free FP-injective R -modules is Gorenstein FP ∞ -injective. [ABSTRACT FROM AUTHOR]

In this paper, we introduce ∗-almost independent rings of Krull type (∗-almost IRKTs) and ∗-almost generalized Krull domains (∗-almost GKDs) in the general theory of almost factoriality, neither of which need be integrally closed. This fills a gap left in [D. D. Anderson and M. Zafrullah, On∗-Semi-Homogeneous Integral Domains, Advances in Commutative Algebra (Springer, Singapore, 2019)]. We characterize them by ∗-almost super-SH domains, where a domain D is called a ∗-almost super-SH domain if every nonzero proper principal ideal of D is a ∗-product of ∗-almost super-homogeneous ideals. We prove that (1) a domain D is a ∗-almost IRKT if and only if D is a ∗-almost super-SH domain, (2) a domain is a ∗-almost GKD if and only if D is a type 1 ∗-almost super-SH domain and (3) a domain D is a ∗-almost IRKT and an AGCD-domain if and only if D is a ∗-afg-SH domain. Further, we characterize them by their integral closures. For example, we prove that a domain D is an almost IRKT if and only if D ⊆ D ¯ is a root extension with D t -linked under D ¯ and D ¯ is an IRKT. Examples are given to illustrate the new concepts. [ABSTRACT FROM AUTHOR]

It is well known in [Absolutely pure modules, Proc. Amer. Math. Soc.26 (1970) 561–566, Theorem 6] that a domain R is a Prüfer domain if and only if every divisible R -module is absolutely pure, where an R -module A is called absolutely pure if Ext R 1 (N , A) = 0 for every finitely presented R -module N. In this paper, we extend this result to Prüfer v -multiplication domains (P v MDs). To do this, comparing with [An Introduction to Homological Algebra, 2nd edn. (Springer, Science+Business Media, LLC, New York, 2009), Theorem 3.69], we firstly give homological characterizations of w -purity, and we introduce the concept of absolutely w -pure modules over commutative rings with zero divisors. Finally, we prove that a domain R is a P v MD if and only if every divisible R -module is absolutely w -pure, and compare absolutely w -purity with absolutely purity by giving an example. [ABSTRACT FROM AUTHOR]

It is well known that a domain is a Prüfer domain if and only if for -modules, relative divisibility implies purity. In this paper, we extend this result to Prüfer -multiplication domains (PMDs). To do this, we introduce the concept of -pure exact sequences over commutative rings with zero divisors, and we prove that a domain is a PMD if and only if for -modules, relative divisibility implies -purity. Also, we compare -purity with purity and relative divisibility by giving examples. [ABSTRACT FROM AUTHOR]

Let be an integral domain, and the set of fractional ideals of . Let a finitely generated ideal with . For a torsion-free -module , define for some . Call a -module if . On , the function is a star-operation of finite character. An integral ideal maximal with respect to being a proper -ideal is a prime ideal called a maximal -ideal. A torsion-free -module is called -flat, if is a flat -module for each , the set of maximal -ideals of . is called a Prüfer -multiplication domain (PMD), if is a valuation ring for each . We characterize -flat modules in a manner similar to the characterization of flat modules, study them when they are rings with and characterize PMDs using them and compare our work with similar work in the literature. [ABSTRACT FROM AUTHOR]

An LPI domain is a domain in which every locally principal ideal is invertible. In this paper, we give a negative answer to a question of Anderson-Zafrullah. We show that if is an LPI domain and is a multiplicatively closed set, then is not necessarily an LPI domain. Concerning a question of Bazzoni, we give a characterization of finite character for a finitely stable LPI-domain. We show that if is a finitely stable LPI-domain, then every nonzero nonunit element of is contained in only finitely many maximal ideals which are in , and is of finite character if and only if each nonzero nonunit element is contained in only finitely many nonstable maximal ideals. A maximal ideal is in provided there is a finitely generated ideal such that is the only maximal ideal containing . [ABSTRACT FROM AUTHOR]