Elementary Factorization in π-Regular Rings

Abstract

This paper extends the results of A. L. Foster (1) on elementary factorization in Boolean-like rings to commutative π-regular rings. After proving some preliminary lemmas we proceed to the partition of the set of non-units of a π-regular ring into irreducible and composite elements. Finally, we prove a number of theorems concerning factorization rings, weakly unique factorization rings, principal ideal rings, etc. The principal result is that a π-regular ring is a weakly unique factorization ring if and only if it is a principal ideal ring.

G-Clean rings

“…Some properties of regular rings and strongly regular has been studied in [1,7,10].A ring is said to be -regular if for every element , there is an element such that ,for some positive integer . A ring is said to be strongly -regular if for every , there is an element such that ,for some positive integer .In many papers concerned -regular and strongly -regular rings, see [3,4,5,6,9,13,14,15,17,20,21,23] for all .…”

Idempotent matrices over a duo ring

“…[5], Lemma 2.1). Let R is duo ring and b, e ∈ R, where e 2 = e. If bR = eR, then b and e is right associates.…”