Associate elements in commutative rings

Abstract

Let $R$ be a commutative ring with identity. For $a,b\in R$, define $a$ and $b$ to be \textit{associates}, denoted $a\sim b$, if $a\mid b$ and $b\mid a$, so $a=rb$ and $b=sa$ for some $r,s\in R$. We are interested in the case where $r$ and $s$ can be taken or must be taken to be non zero-divisors or units. We study rings, $R$, called \textit{strongly regular associate}, that have the property that, whenever $a\sim b$ for $a,b\in R$, then there exist non zero-divisors $r,s\in R$ with $a=rb$ and $b=sa$ and rings $R$, called \textit{weakly pr\'{e}simplifiable}, that have the property that, for nonzero $a,b\in R$ with $a\sim b$, whenever $a=rb$ and $b=sa$, then $r$ and $s$ must be non zero-divisors.