On countably $\Sigma$-C2 rings

TL;DR

This paper characterizes right countably Sigma-C2 rings, showing they are equivalent to rings with right finitistic projective dimension zero, connecting module-theoretic properties with ring-theoretic conditions.

Contribution

It provides a new characterization of right countably Sigma-C2 rings via equivalences involving matrix rings and projective modules, linking to Bass's finitistic projective dimension.

Findings

Equivalence of conditions for right countably Sigma-C2 rings.

Connection to right finitistic projective dimension zero.

Characterization involving matrix rings and module properties.

Abstract

Let $R$ be a ring. $R$ is called a right countably $\Sigma$-C2 ring if every countable direct sum copies of $R_{R}$ is a C2 module. The following are equivalent for a ring $R$: (1) $R$ is a right countably $\Sigma$-C2 ring. (2) The column finite matrix ring $\mathbb{C}\mathbb{F}\mathbb{M}_{\mathbb{N}}(R)$ is a right C2 (or C3) ring. (3) Every countable direct sum copies of $R_{R}$ is a C3 module. (4) Every projective right $R$-module is a C2 (or C3) module. (5) $R$ is a right perfect ring and every finite direct sum copies of $R_{R}$ is a C2 (or C3) module. This shows that right countably $\Sigma$-C2 rings are just the rings whose right finitistic projective dimension r$FPD(R)$=sup\{$Pd_{R}(M)|$ $M$ is a right $R$-module with $Pd_{R}(M)<\infty$\}=0, which were introduced by Hyman Bass in 1960.