We first introduce the notion of an Auslander-Reiten sequence in a Krull-Schmidt category. This unifies the notion of an almost split sequence in an abelian category and that of an Auslander-Reiten triangle in a triangulated category. We then define the Auslander-Reiten quiver of a Krull-Schmidt category and describe the shapes of its semi-stable components. The main result generalizes those fo...

The principle “Every result in classical homological algebra should have a counterpart in Gorenstein homological algebra” was given by Henrik Holm. There is a remarkable body of evidence supporting this claim. Perhaps one of the most glaring exceptions is provided by the fact that tensor products of Gorenstein projective modules need not be Gorenstein projective, even over Gorenstein rings. So ...

Extending work of Puninski, Puninskaya and Toffalori in [PPT07], we show that if V is an effectively given valuation domain then the theory of all V -modules is decidable if and only if there exists an algorithm which, given a, b ∈ V , answers whether a ∈ rad(bV ). This was conjectured in [PPT07] for valuation domains with dense value group where it was proved for valuation domains with dense a...

The main goal of this paper is to introduce and study new classes partial orderings on a set with Krull dimension at most 1. We show that these are related door, submaximal Whyburn T0-spaces.

The algebra Sn of one-sided inverses of a polynomial algebra Pn in n variables is obtained from Pn by adding commuting, left (but not two-sided) inverses of the canonical generators of the algebra Pn. The algebra Sn is a noncommutative, non-Noetherian algebra of classical Krull dimension 2n and of global dimension n and is not a domain. If the ground field K has characteristic zero then the alg...

In this paper, we consider flat epimorphisms of commutative rings [Formula: see text] such that, for every ideal which text], the quotient ring is semilocal Krull dimension zero. Under these assumptions, show that projective R-module U does not exceed 1. We also describe Geigle–Lenzing perpendicular subcategory in text]. Assuming additionally and all are perfect, R-modules U-strongly flat. Thus...