AF-domains and their generalizations

TL;DR

This paper explores the dimension theory of tensor products of algebras over a field, introducing generalized AF-domains and computing their Krull dimensions, extending previous results in the field.

Contribution

It introduces the concept of GAF-domains, shows that polynomial extensions of AF-domains are GAF-domains, and generalizes the Krull dimension formula for tensor products.

Findings

Polynomial rings over AF-domains are GAF-domains.

Krull dimension of tensor products is explicitly computed.

Extension of Wadsworth's main theorem to broader classes.

Abstract

In this paper, we are concerned with the study of the dimension theory of tensor products of algebras over a field $k$. We introduce and investigate the notion of generalized AF-domain (GAF-domain for short) and prove that any $k$-algebra $A$ such that the polynomial ring in one variable $A[X]$ is an AF-domain is in fact a GAF-domain, in particular any AF-domain is a GAF-domain. Moreover, we compute the Krull dimension of $A\otimes_kB$ for any $k$-algebra $A$ such that $A[X]$ is an AF-domain and any $k$-algebra $B$ generalizing the main theorem of Wadsworth in [16].