Abstract
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.
ACKNOWLEDGMENT
The authors wish to express their sincere thanks to Professor R. Wisbauer and to the referee for their valuable suggestions. The authors were supported by National Natural Science Foundation of China (10171082), TRAPOYT, and the Cultivation Fund of the Key Scientific and Technical Innovation Project, Ministry of Education of China.
Notes
Communicated by R. Wisbauer.