RING ENDOMORPHISMS WITH THE REVERSIBLE CONDITION

Title & Authors
RING ENDOMORPHISMS WITH THE REVERSIBLE CONDITION
Baser, Muhittin; Kaynarca, Fatma; Kwak, Tai-Keun;

Abstract
P. M. Cohn called a ring R reversible if whenever ab = 0, then ba = 0 for a, $\small{b\;{\in}\;R}$. Commutative rings and reduced rings are reversible. In this paper, we extend the reversible condition of a ring as follows: Let R be a ring and $\small{\alpha}$ an endomorphism of R, we say that R is right (resp., left) $\small{\alpha}$-shifting if whenever $\small{a{\alpha}(b)\;=\;0}$ (resp., $\small{\alpha{a)b\;=\;0}$) for a, $\small{b\;{\in}\;R}$, $\small{b{\alpha}{a)\;=\;0}$ (resp., $\small{\alpha(b)a\;=\;0}$); and the ring R is called $\small{\alpha}$-shifting if it is both left and right $\small{\alpha}$-shifting. We investigate characterizations of $\small{\alpha}$-shifting rings and their related properties, including the trivial extension, Jordan extension and Dorroh extension. In particular, it is shown that for an automorphism $\small{\alpha}$ of a ring R, R is right (resp., left) $\small{\alpha}$-shifting if and only if Q(R) is right (resp., left) $\small{\bar{\alpha}}$-shifting, whenever there exists the classical right quotient ring Q(R) of R.
Keywords
ring endomorphism;reduced ring;reversible ring;trivial extension;classical right quotient ring;
Language
English
Cited by
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