On quasi-rigid ideals and rings
Bull. Korean Math. Soc. 2010 Vol. 47, No. 2, 385-399
https://doi.org/10.4134/BKMS.2010.47.2.385
Printed March 1, 2010
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
MSC numbers : 16N60, 16S36, 16W60
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