On p.p.-rings which are reduced

Throughout the paper, all rings are associative rings with identity 1. The set of all idempotents of a ring R is denoted by E(R). Also, for a subset X ⊆ R, we denote the right [resp., left] annihilator of X by r(X) [resp., (X)]. We call a ring R a left p.p.-ring [3], in brevity, an l.p.p.-ring, if every principal left ideal of R, regarded as a left R-module, is projective. Dually, we may define the right p.p.-rings (r.p.p.-rings). We call a ring R a p.p.-ring if R is both an l.p.p.and r.p.p.-ring. It can be easily observed that the class of p.p.-rings contains the classes of regular (von Neumann) rings, hereditary rings, Baer rings, and semihereditary rings as its proper subclasses. In the literature, p.p.-rings have been extensively studied by many authors and many interesting results have been obtained (see [1–7]). It is noteworthy that the definition of p.p.-rings can also be extended to semigroups. We now call a ring R reduced if it contains no nonzero nilpotent elements. Obviously, the left annihilator (X) of X in a reduced ring R is always a two-sided ideal of R. Moreover, if R is a reduced ring, then e f = 0 if and only if f e = 0 for any nonzero idempotents e, f ∈ R. Reduced rings with the maximum condition on annihilator were first studied by Cornish and Stewart [2]. By using the concept of annihilator and reduced ring, Fraser and Nicholson [3] showed that a ring R is a reduced p.p.-ring if and only if R is a (left, right) p.p.-ring in which every idempotent is central. In this paper, we will prove that a p.p.-ring R is reduced if and only if R contains no subrings which are isomorphic to the matrix rings UTM2(Z) or UTM2(Zp). Thus, our