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In this paper, we construct some non-integrally closed domains in Gorenstein multiplicative ideal theory. For example, we show that there exists a Gorenstein Prüfer domain which is neither Gorenstein Dedekind nor Prüfer, and there exists a Gorenstein Krull domain which is neither Gorenstein Dedekind nor Krull. Also, we construct a non-integrally closed non-coherent domain in which all Gorenstein projective (resp., injective, flat) modules are projective (resp., injective, flat). [ABSTRACT FROM AUTHOR]

Let D be an integral domain. Then D is an almost valuation (AV)-domain if for a , b ∈ D ∖ { 0 } there exists a natural number n with a n | b n or b n | a n. AV-domains are closely related to valuation domains, for example, D is an AV-domain if and only if the integral closure D ¯ is a valuation domain and D ⊆ D ¯ is a root extension. In this note, we explore various generalizations of DVRs (which we might call almost DVRs) such as Noetherian AV-domains, AV-domains with D ¯ a DVR, and quasilocal and local API-domains (i.e., for { a α } α ∈ Λ ⊆ D , there exists an n with ( { a α n } α ∈ Λ) principal). The structure of complete local AV-domains and API-domains is determined. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity. An R-module E is said to be weak injective if for any super finitely presented R-module N. In this article, for a domain R with quotient field , it is characterized when K/R as an R-module is weak injective. Also, a couple of characterizations of generalized coherent domains of self-weak injective dimension one are given. As corollaries, we characterize Gorenstein Dedekind (resp., Gorenstein Prüfer) domains. [ABSTRACT FROM AUTHOR]