Gelfand-Kirillov dimension for rings

TL;DR

This paper extends the understanding of Gelfand-Kirillov dimension to various algebraic constructions over commutative domains, providing explicit computations and broadening its applicability.

Contribution

It offers complete proofs for the Gelfand-Kirillov dimension of polynomial rings, matrix rings, localizations, and skew PBW extensions over commutative domains, including modules and transcendence degree.

Findings

Computed Gelfand-Kirillov dimension for polynomial and matrix rings

Extended results to localizations and skew PBW extensions

Applicable to algebras over rings of integers

Abstract

The classical Gelfand-Kirillov dimension for algebras over fields has been extended recently by J. Bell and J.J Zhang to algebras over commutative domains. However, the behavior of this new notion has not been enough investigated for the principal algebraic constructions as polynomial rings, matrix rings, localizations, filtered-graded rings, skew $PBW$ extensions, etc. In this paper, we present complete proofs of the computation of this more general dimension for the mentioned algebraic constructions for algebras over commutative domains. The Gelfand-Kirillov dimension for modules and the Gelfand-Kirillov transcendence degree will be also considered. The obtained results can be applied in particular to algebras over the ring of integers, i.e, to arbitrary rings.