We introduce weak Armendariz rings which are a generalization of semicommutative rings and Armendariz rings, and investigate their properties. Moreover, we prove that a ring R is weak Armendariz if and only if for any n , the n -by- n upper triangular matrix ring T-n(R) is weak Armendariz. If R is semicommutative, then it is proven that the polynomial ring R [x] over R and the ring R [x =]/(x(n)), where (x(n)) is the ideal generated by x(n) and n is a positive integer, are weak Armendariz.
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