Abstract
As a popularization of weakly π-regular rings, we tender the connotation of W
P
Wπ-regular rings, that is if for each 𝔞 ∈ Ɉ(𝔑), there exist a natural number 𝔫 such that 𝔞𝔫 ∈ 𝔞𝔫 𝔑 𝔞𝔫 𝔑 (𝔞𝔫 ∈ 𝔑 𝔞𝔫 𝔑 𝔞𝔫). In this treatise, numerous properties of this sort of rings are discussed, some important results are secured. Using the connotation of
P
Wπ-regular rings. It is show that :
1- Let 𝔑 be a right
P
Wπ-regular ring and ℵɈ-rings with 𝔞𝔫𝔑 = 𝔑𝔞𝔫
for every 𝔞 ∈ Ɉ(𝔑) and for at least one of a natural number. Then Ɉ(𝔑) = ℵ𝔑.
2- Let 𝔑 a right
P
Wπ-regular ring and 𝔞𝔑 = 𝔑𝔞 for each ∈Ɉ(𝔑). Then 𝔑 is right
P
.𝔗-ring.
3 Let 𝔑 be a ring with ɍ(𝔞) ⊆ ɭ(𝔞), for each ∈ Ɉ (𝔑). If any of the next conditions are hold, then 𝔑 is
P
Wπ-regular rings :
i – Every maximal right ideal of 𝔑 is a right annihilator and right Ɉ
PP
-ring.
ii – any simple singular right 𝔑-module is Ɉ-injective and 𝔑 is semi prime.