Rocky Mountain Journal of Mathematics

Associate elements in commutative rings

D.D. Anderson and Sangmin Chun

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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.

Article information

Rocky Mountain J. Math., Volume 44, Number 3 (2014), 717-731.

First available in Project Euclid: 28 September 2014

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Zentralblatt MATH identifier

Primary: 13A05: Divisibility; factorizations [See also 13F15] 13A15: Ideals; multiplicative ideal theory 13F99: None of the above, but in this section


Anderson, D.D.; Chun, Sangmin. Associate elements in commutative rings. Rocky Mountain J. Math. 44 (2014), no. 3, 717--731. doi:10.1216/RMJ-2014-44-3-717.

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