A note on minimal prime ideals
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Let $R$ be a strongly $2$-primal ring and $I$ a proper ideal of $R$. Then there are only finitely many prime ideals minimal over $I$ if and only if for every prime ideal $P$ minimal over $I$, the ideal $P/\sqrt{I}$ of $R/\sqrt{I}$ is finitely generated if and only if the ring $R/\sqrt{I}$ satisfies the \emph{ACC} on right annihilators. This result extends ``D. D. Anderson, A note on minimal prime ideals, \emph{Proc. Amer. Math. Soc.} 122 (1994), no. 1, 13--14." to large classes of noncommutative rings. It is also shown that, a $2$-primal ring $R$ only has finitely many minimal prime ideals if each minimal prime ideal of $R$ is finitely generated. Examples are provided to illustrate our results.
Abstract
Let $R$ be a strongly $2$-primal ring and $I$ a proper ideal of $R$. Then there are only finitely many prime ideals minimal over $I$ if and only if for every prime ideal $P$ minimal over $I$, the ideal $P/\sqrt{I}$ of $R/\sqrt{I}$ is finitely generated if and only if the ring $R/\sqrt{I}$ satisfies the \emph{ACC} on right annihilators. This result extends ``D. D. Anderson, A note on minimal prime ideals, \emph{Proc. Amer. Math. Soc.} 122 (1994), no. 1, 13--14." to large classes of noncommutative rings. It is also shown that, a $2$-primal ring $R$ only has finitely many minimal prime ideals if each minimal prime ideal of $R$ is finitely generated. Examples are provided to illustrate our results.
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