International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 34694, 5 pages
doi:10.1155/IJMMS/2006/34694

On p.p.-rings which are reduced

Xiaojiang Guo1 and K. P. Shum2

1Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330027, China
2Faculty of Science, The Chinese University of Hong Kong, Shatin, Hong Kong

Received 16 March 2006; Accepted 19 March 2006

Copyright © 2006 Xiaojiang Guo and K. P. Shum. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Denote the 2×2 upper triangular matrix rings over ℤ and ℤp by UTM2(ℤ) and UTM2(ℤp), respectively. We prove that if a ring R is a p.p.-ring, then R is reduced if and only if R does not contain any subrings isomorphic to UTM2(ℤ) or UTM2(ℤp). Other conditions for a p.p.-ring to be reduced are also given. Our results strengthen and extend the results of Fraser and Nicholson on r.p.p.-rings.