Volume 7, Issue 3 (Vol.7, No.3, 2021)
mmr 2021, 7(3): 585-590 |
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sahebi S. Semi-Armendariz and Semi-McCoy rings. mmr 2021; 7 (3) :585-590
URL: http://mmr.khu.ac.ir/article-1-2915-en.html
URL: http://mmr.khu.ac.ir/article-1-2915-en.html
, sahebi@iauctb.ac.ir
Abstract: (1016 Views)
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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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../files/site1/files/%D8%B5%D8%A7%D8%AD%D8%A8%DB%8C-%DA%86%DA%A9%DB%8C%D8%AF%D9%87-_%D8%A7%D9%86%DA%AF%D9%84%DB%8C%D8%B3%DB%8C(1).pdf
Type of Study: S |
Subject:
alg
Received: 2019/02/19 | Revised: 2022/05/7 | Accepted: 2020/03/1 | Published: 2021/12/1 | ePublished: 2021/12/1
Received: 2019/02/19 | Revised: 2022/05/7 | Accepted: 2020/03/1 | Published: 2021/12/1 | ePublished: 2021/12/1
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