Semi-Armendariz and Semi-McCoy rings

Author
shervin sahebi
10.52547/mmr.7.3.585
Abstract
We introduce the notion of Semi-Armendariz (resp. Semi-McCoy) rings, which are a subclass of J-Armendariz (resp. J-McCoy rings) and investigate their properties. A ring R is called Semi-Armendariz (Semi-McCoy) if is Armendariz (McCoy). As special case, we show that the class of Semi-Armendariz (resp. Semi-McCoy) rings lies properly between the class of one-sided quasi-duo rings and the class of J-Armendariz (resp. J-McCoy) rings. We show that a ring R is Semi-Armendariz (resp. Semi-McCoy) iff R[[x]] is Semi-Armendariz (resp. Semi-McCoy) iff for any idempotent , eRe is Semi-Armendariz (resp. Semi-McCoy) iff the n-by-n upper triangular matrix ring Tn(R) is Semi-Armendariz (resp. Semi-McCoy). But, by an example we show that for a ring R and n>1, is not necessarily Semi-Armendariz (Semi-McCoy) and so R is not Morita invariant. At last, we prove that for an automorphism a ring R is Semi-Armendariz (resp. Semi-McCoy) iff the Jordan structure of R ( is Semi-Armendariz (resp. Semi-McCoy) and so we identify the Jacobson radical of A.

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Keywords
: J-Armendariz ring
J-McCoy ring
Quasi duo-ring
Jacobson radical
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Mathematical Research
Volume 7, Issue 3 - Serial Number 18
Autumn 2021
Pages 585-590