A characterization of factorial rings
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초록
Let $R$ be a commutative ring. It is proved in this note that if every (regular) prime $w$-ideal of $R$ is principal, then every (regular) $w$-ideal of $R$ is also principal. It is also proved that if every regularly principal ideal $I$ (regular elements in $I$ are contained in a principal ideal which is generated by a regular element in $I$) of $R$ is principal, for example, zero-divisors in $R$ are nilpotent, then $R$ is a factorial ring if and only if every regular $w$-ideal of $R$ is principal.
Abstract
Let $R$ be a commutative ring. It is proved in this note that if every (regular) prime $w$-ideal of $R$ is principal, then every (regular) $w$-ideal of $R$ is also principal. It is also proved that if every regularly principal ideal $I$ (regular elements in $I$ are contained in a principal ideal which is generated by a regular element in $I$) of $R$ is principal, for example, zero-divisors in $R$ are nilpotent, then $R$ is a factorial ring if and only if every regular $w$-ideal of $R$ is principal.