Radical preservation and the finitistic dimension

TL;DR

This paper introduces radical preservation in ring homomorphisms and demonstrates its significance in reflecting the finiteness of various finitistic dimensions, with applications to bound quiver algebras and semiprimary rings.

Contribution

It defines radical preservation and proves its role in relating finiteness properties of finitistic dimensions, including new classes of algebras with finite finitistic dimension.

Findings

Radical-preserving homomorphisms reflect finiteness of finitistic dimensions.

Bound quiver algebras with quasi-uniform Loewy length have finite finitistic dimension.

Constructed examples show finiteness results are not implied by existing literature.

Abstract

We introduce the notion of radical preservation and prove that a radical-preserving homomorphism of left artinian rings of finite projective dimension with superfluous kernel reflects the finiteness of the little finitistic, big finitistic and global dimension. As an application, we prove that every bound quiver algebra with quasi-uniform Loewy length, a class of algebras introduced in this paper, has finite (big) finitistic dimension. The same result holds more generally in the context of semiprimary rings. Moreover, we construct an explicit family of such finite dimensional algebras where the finiteness of their big finitistic dimension does not follow from existing results in the literature.