Semi-Armendariz and Semi-McCoy rings

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. ./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

Keywords

• : j-armendariz ring
• j-mccoy ring
• quasi duo-ring
• jacobson radical.