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 T
n(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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