A note on near completely prime ideal rings over σ(*)-rings

Abstract

Let $R$ be a ring and $\sigma$ an endomorphism of $R$. Recall that $R$ is said to be a $\sigma(*)$-ring if $a\sigma(a) \in P(R)$ for $a \in R$, where $P(R)$ is the prime radical of $R$. We also recall that a ring $R$ is said to be a completely prime ideal ring (CPI-ring) if every prime ideal of $R$ is completely prime. We say that a ring $R$ is a near completely prime ideal ring (NCPI-ring) if every minimal prime ideal of $R$ is completely prime Bhat [6].

In this paper we give a relation between $\sigma(*)$-ring and near completely prime ideal ring and also proved that if $R$ is a Noetherian ring and $\sigma$ an endomorphism of $R$ such that $R$ is $\sigma(*)$-ring then $S(R) = R[x; \sigma]$ is a Noetherian near completely prime ideal ring.