ON QUASI-RIGID IDEALS AND RINGS

Title & Authors
ON QUASI-RIGID IDEALS AND RINGS
Hong, Chan-Yong; Kim, Nam-Kyun; Kwak, Tai-Keun;

Abstract
Let $\small{\sigma}$ be an endomorphism and I a $\small{\sigma}$-ideal of a ring R. Pearson and Stephenson called I a $\small{\sigma}$-semiprime ideal if whenever A is an ideal of R and m is an integer such that $\small{A{\sigma}^t(A)\;{\subseteq}\;I}$ for all $\small{t\;{\geq}\;m}$, then $\small{A\;{\subseteq}\;I}$, where $\small{\sigma}$ is an automorphism, and Hong et al. called I a $\small{\sigma}$-rigid ideal if $\small{a{\sigma}(a)\;{\in}\;I}$ implies a $\small{a\;{\in}\;I}$ for $\small{a\;{\in}\;R}$. Notice that R is called a $\small{\sigma}$-semiprime ring (resp., a $\small{\sigma}$-rigid ring) if the zero ideal of R is a $\small{\sigma}$-semiprime ideal (resp., a $\small{\sigma}$-rigid ideal). Every $\small{\sigma}$-rigid ideal is a $\small{\sigma}$-semiprime ideal for an automorphism $\small{\sigma}$, but the converse does not hold, in general. We, in this paper, introduce the quasi $\small{\sigma}$-rigidness of ideals and rings for an automorphism $\small{\sigma}$ which is in between the $\small{\sigma}$-rigidness and the $\small{\sigma}$-semiprimeness, and study their related properties. A number of connections between the quasi $\small{\sigma}$-rigidness of a ring R and one of the Ore extension $\small{R[x;\;{\sigma},\;{\delta}]}$ of R are also investigated. In particular, R is a (principally) quasi-Baer ring if and only if $\small{R[x;\;{\sigma},\;{\delta}]}$ is a (principally) quasi-Baer ring, when R is a quasi $\small{\sigma}$-rigid ring.
Keywords
endomorphism;rigidness;semiprimeness;Ore extension;(principally) quasi-Baer ring;
Language
English
Cited by
1.
QUASI-ARMENDARIZ PROPERTY FOR SKEW POLYNOMIAL RINGS,;;

대한수학회논문집, 2011. vol.26. 4, pp.557-573
1.
QUASI-ARMENDARIZ PROPERTY FOR SKEW POLYNOMIAL RINGS, Communications of the Korean Mathematical Society, 2011, 26, 4, 557
2.
Special properties of the ring Sn(R), Journal of Algebra and Its Applications, 2016, 1750212
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