$\tau$-Atomicity and Quotients of Size Four

Abstract

Given a ring $R$, an ideal $I$ of $R$,  and an element $a\in I$,  we say $a=\lambda b_1\cdots b_k$ is a $\tau_I$-factorization of $a$ if $\lambda$ is any unit and $b_1\equiv\cdots\equiv b_k\pmod{I}$.  In this paper, we investigate the $\tau_I$-atomicity of PIDs with ideals where $R/I$ has size four.