Chan Yong Hong, Nam Kyun Kim, and Tai Keun Kwak Kyung Hee University, Hanbat National University, and Daejin University
Abstract : Let $\sigma$ be an endomorphism and $I$ a $\sigma$-ideal of a ring $R$. Pearson and Stephenson called $I$ a $\sigma$-semiprime ideal if whenever $A$ is an ideal of $R$ and $m$ is an integer such that $A\sigma^t(A) \subseteq I$ for all $t\geq m$, then $A \subseteq I$, where $\sigma$ is an automorphism, and Hong et al. called $I$ a $\sigma$-rigid ideal if $a\sigma(a)\in I$ implies $a\in I$ for $a\in R$. Notice that $R$ is called a $\sigma$-semiprime ring (resp., a $\sigma$-rigid ring) if the zero ideal of $R$ is a $\sigma$-semiprime ideal (resp., a $\sigma$-rigid ideal). Every $\sigma$-rigid ideal is a $\sigma$-semiprime ideal for an automorphism $\sigma$, but the converse does not hold, in general. We, in this paper, introduce the quasi $\sigma$-rigidness of ideals and rings for an automorphism $\sigma$ which is in between the $\sigma$-rigidness and the $\sigma$-semiprimeness, and study their related properties. A number of connections between the quasi $\sigma$-rigidness of a ring $R$ and one of the Ore extension $R[x;\sigma,\delta]$ of $R$ are also investigated. In particular, $R$ is a (principally) quasi-Baer ring if and only if $R[x;\sigma,\delta]$ is a (principally) quasi-Baer ring, when $R$ is a quasi $\sigma$-rigid ring.
Keywords : endomorphism, rigidness, semiprimeness, Ore extension, (principally) quasi-Baer ring