P-ideals and PMP-ideals in Commutative rings
DOI:
https://doi.org/10.30495/jme.v10i0.422Keywords:
P-ideal, PMP-ideal, pure ideal, von Neumann regular ideal, P-space.Abstract
Recently, P-ideals studied in ${\rm C}(X)$ by some authors. In this article we investigate {\rm P}-ideals and a new concept as PMP-ideals in commutative rings. We show that $I$ is a {\rm P}-ideal (resp.,{\rm PMP}-ideal) in $R$ if and only if every prime ideal of $R$ which
does not contain $I$ is a maximal (resp., minimal prime) ideal of
$R$. Also, we characterize largest {\rm P}-ideals (resp., PMP-ideals) in commutative rings, specially in ${\rm C}(X)$. Furthermore, we study relation between these ideals and pure ideals.
%\end{abstract}
Finally we prove that ${\rm C}(X)$ is a von Neumann regular ring if and only if every pure ideal of it is P-ideal.
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