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Let R be a commutative Noetherian ring. In this article, we provide some new criteria for a semidualizing module to be dualizing in terms of special homological properties of module categories. The purpose of this article is twofold: first, it aims at improving Christensen's and Takahashi et al.'s characterizations of dualizing modules; secondly, while applying these criteria to the ring itself, we not only recover some results of Jenda and Xu, respectively, but also obtain a new characterization of Gorenstein rings. [ABSTRACT FROM PUBLISHER]

Let k be an algebraically closed field of characteristic 0, and let A = k[x, y]/(f) be a quasi-homogeneous plane curve. We show that for any graded torsion free A-module M, there exists a natural graded integrable connection, i.e., a graded A-linear homomorphism ∇: Derk(A) → Endk(M), that satisfies the derivation property and preserves the Lie product. In particular, a torsion free module N over the complete local ring [image omitted] admits a natural integrable connection if A is a simple curve singularity, or if A is irreducible and N is a gradable module. [ABSTRACT FROM AUTHOR]

Recently Jain, López-Permouth and Syed classified indecomposable non-local right continuous right π-rings. In the present article we describe the structure of right continuous right π-rings. [ABSTRACT FROM AUTHOR]