The small finitistic dimensions of commutative rings, II

TL;DR

This paper studies the small finitistic dimensions of various ring constructions, establishing bounds related to Krull dimension and providing examples with infinite small finitistic dimension.

Contribution

It extends the understanding of small finitistic dimensions to polynomial, power series, trivial extensions, and amalgamations of rings, and relates these to Krull dimension.

Findings

Small finitistic dimension is bounded above by Krull dimension.

Determined small finitistic dimensions for polynomial and power series rings.

Constructed examples with infinite small finitistic dimension.

Abstract

The small finitistic dimension $\fPD(R)$ of a ring $R$ is defined to be the supremum of projective dimensions of $R$-modules with finite projective resolutions. In this paper, we investigate the small finitistic dimensions of four types of ring constructions: polynomial rings, formal power series rings, trivial extensions and amalgamations. Besides, we show the small finitistic dimensions of a ring is less than or equal to its Krull dimension. We also give a total ring of quotients with infinite small finitistic dimension.