Armendariz and Reduced Rings

Abstract

A ring R is called Armendariz if, whenever in R[x], a _i b _j  = 0 for all i and j. In this paper, some “relatively maximal” Armendariz subrings of matrix rings are identified, and a necessary and sufficient condition for a trivial extension to be Armendariz is obtained. Consequently, new families of Armendariz rings are presented.

Key Words:

• Armendariz ring; Matrix ring; Reduced ring; Trivial extension

Mathematics Subject Classification:

• 16S36; 16S50; 16U99

Acknowledgments

The second author is grateful to the kind hospitality of National Taiwan University when part of the work was done during his visit there. The research is supported by NSERC, Grant OGP 0194196, of Canada and NSC of R.O.C. (Taiwan).

Notes

^#Communicated by R. Wisbauer.