SKEW POLYNOMIAL RINGS OVER σ-QUASI-BAER AND σ-PRINCIPALLY QUASI-BAER RINGS

Title & Authors
SKEW POLYNOMIAL RINGS OVER σ-QUASI-BAER AND σ-PRINCIPALLY QUASI-BAER RINGS
HAN JUNCHEOL;

Abstract
Let R be a ring R and $\small{{\sigma}}$ be an endomorphism of R. R is called $\small{{\sigma}}$-rigid (resp. reduced) if $\small{a{\sigma}r(a) = 0 (resp{\cdot}a^2 = 0)}$ for any $\small{a{\in}R}$ implies a = 0. An ideal I of R is called a $\small{{\sigma}}$-ideal if $\small{{\sigma}(I){\subseteq}I}$. R is called $\small{{\sigma}}$-quasi-Baer (resp. right (or left) $\small{{\sigma}}$-p.q.-Baer) if the right annihilator of every $\small{{\sigma}}$-ideal (resp. right (or left) principal $\small{{\sigma}}$-ideal) of R is generated by an idempotent of R. In this paper, a skew polynomial ring A = R[$\small{x;{\sigma}}$] of a ring R is investigated as follows: For a $\small{{\sigma}}$-rigid ring R, (1) R is $\small{{\sigma}}$-quasi-Baer if and only if A is quasi-Baer if and only if A is $\small{\={\sigma}}$-quasi-Baer for every extended endomorphism $\small{\={\sigma}}$ on A of $\small{{\sigma}}$ (2) R is right $\small{{\sigma}}$-p.q.-Baer if and only if R is $\small{{\sigma}}$-p.q.-Baer if and only if A is right p.q.-Baer if and only if A is p.q.-Baer if and only if A is $\small{\={\sigma}}$-p.q.-Baer if and only if A is right $\small{\={\sigma}}$-p.q.-Baer for every extended endomorphism $\small{\={\sigma}}$ on A of $\small{{\sigma}}$.
Keywords
$\small{\sigma}$-rigid ring;$\small{\sigma}$-Baer ring;$\small{\sigma}$-quasi-Baer ring;$\small{\sigma}$-p.q.-Baer ring;$\small{\sigma}$-p.p. ring;skew polynomial ring;
Language
English
Cited by
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