Rickart gamma rings

Abstract

We introduce the concepts of Rickart gamma ring as a generalized of a Rickart ring and Baer gamma ring. Also, For an Γ —ring R, we show that: (1) R is a right Rickart if and only if it is right p.p.. (2) R is a Baer, quasi-Baer if and only if R is Rickart, p.q. Baer (respectively) and R = {eΓR|e 2 = e ∈ R|, under inclusion, is complete lattice. (3) R is prime if and only if it is quasi-(or right p.q.) Baer and semicentral reduced. (4) If R is Rickart then: (a) R is both right and left nonsingular. (b) R has no nonzero central nilpotent elements. (c) The image isomorphic of R is Rickart. (d) If R is reduced then the idempotent element which is generated the right annihilator of any element in is R unique. (5) The direct product Π i∈I Ri of Γ —rings is Rickart, quasi-(right p.q. ) Baer iff Ri is Rickart quasi-(right p.q.) Baer for all i ∈ I, respectively. (6) The corner and the center of Quasi- Baer, p.q. Baer and Rickart are Quasi-Baer and Baer, p.q. and Rickart and, by conditions, Rickart and Rickart, respectively. (7) We give some conditions to show that (a) If R is Rickart, Quasi-Baer then R Baer and p.q.- Baer, Rickart, respectively. (b) If R is Rickart, then the following statements of R are equivalent: reduced, abelian, idempotents of R commute, the set of idempotents is closed under multiplication, R is commutative at 0, RFI(x) = LFI(x) for every x ∈ R and rR (x) = lR (x) for all x ∈ R.