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.

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...

Hanna Neumann asked whether it was possible for two non-isomorphic residually nilpotent finitely generated (fg) groups, one of them free, to share the lower central sequence. G. Baumslag answered the question in the affirmative and thus gave rise to parafree groups. A group G is termed parafree of rank n if it is residually nilpotent and shares the same lower central sequence with a free group ...

Let Z be the center of a nonnoetherian dimer algebra on torus. Although itself is also nonnoetherian, we show that it has Krull dimension 3, and locally noetherian an open dense set Max . Furthermore, reduced / nil depicted by Gorenstein singularity, contains precisely one closed point positive geometric dimension.