Reduced p.p.-rings without identity

Throughout this paper, the ring R is not necessarily with an identity. We denote the set of all idempotents of R by E(R). Also, for a subset X ⊆ R, we denote the right (resp., left) annihilator of X in R by annr(X) (resp., ann (X)). Now, according to Fraser and Nicholson in [5], we call a ring R a left p.p.-ring, in brevity, l.p.p.-ring, if for all x ∈ R, there exists an idempotent e such that ann (x) = ann (e) and ex = x. Dually, we may define a right p.p.-ring. Naturally, we call a ring R a p.p.-ring if it is both an l.p.p.-ring and an r.p.p.-ring. Clearly, if the ring R has an identity, the above (left; right) p.p.-rings coincide with the (left; right) p.p.-rings discussed in [6]. It can be observed that the class of p.p.-rings contains the classes of regular (von Neumann) rings, hereditary rings, Baer rings, and semi-hereditary rings as its proper subclasses. In the literature, p.p.-rings have already been studied by many authors (see [1, 2, 5–9, 11]). It is noteworthy that the definition of p.p.-rings has been extended to semigroups; in particular, Fountain [4] has introduced the concept of abundant semigroups which are both l.p.p.and r.p.p.semigroups. Similar to p.p.-rings, the class of abundant semigroups contains the class of regular semigroups as its proper subclass. An r.p.p.-semigroup in which every idempotent is central is called a C-r.p.p.-semigroup. In 1977 Fountain [3] first proved that a C-r.p.p.-monoid can be expressed as a strong semilattice of left cancellative monoids. This shows that a C-r.p.p.-monoid does not contain any nonzero nilpotent element and hence it is a reduced semigroup. On the other hand, Cornish and Stewart [2] called a ring R reduced if it contains no nonzero nilpotent elements. Obviously, the left annihilator ann (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