On p.p.-rings which are reduced

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.