Gaussian trivial ring extensions and fqp-rings

TL;DR

This paper characterizes when trivial ring extensions are arithmetical, Gaussian, or B{é}zout, introduces fqf-rings as a weaker class than fqp-rings, and analyzes their weak and global dimensions.

Contribution

It provides necessary and sufficient conditions for trivial ring extensions to be arithmetical or Gaussian and introduces fqf-rings, expanding understanding of their properties and dimensions.

Findings

Trivial ring extensions can be arithmetical or Gaussian under specific conditions.

Fqf-rings have finitistic weak dimension 0, 1, or 2.

Global weak dimension of fqf-rings is 0, 1, or infinite.

Abstract

Let A be a commutative ring and E a non-zero A-module. Necessary and sufficient conditions are given for the trivial ring extension R of A by E to be either arithmetical or Gaussian. The possibility for R to be B{\'e}zout is also studied, but a response is only given in the case where pSpec A (a quotient space of Spec} A) is totally disconnected. Trivial ring extensions which are fqp-rings are characterized only in the local case. To get a general result we intoduce the class of fqf-rings satisfying a weaker property than fqp-ring. Moreover, it is proven that the finitistic weak dimension of a fqf-ring is 0, 1 or 2 and its global weak dimension is 0, 1 or infinite.