The $S$-global dimensions of commutative rings

TL;DR

This paper introduces and studies the $S$-global dimension of commutative rings, extending classical homological dimensions by incorporating a multiplicative subset, and explores its behavior in factor and polynomial rings.

Contribution

It defines the $S$-global dimension for commutative rings and investigates its properties, including behavior in factor rings and polynomial extensions.

Findings

Defined $S$-projective and $S$-injective dimensions.

Established properties of $S$-global dimension.

Analyzed $S$-global dimension in factor and polynomial rings.

Abstract

Let $R$ be a commutative ring with identity and $S$ a multiplicative subset of $R$. First, we introduce and study the $S$-projective dimensions and $S$-injective dimensions of $R$-modules, and then explore the $S$-global dimension $S$-gl.dim$(R)$ of a commutative ring $R$ which is defined to be the supremum of $S$-projective dimensions of all $R$-modules. Finally, we investigated the $S$-global dimension of factor rings and polynomial rings.