Reflexive Idempotent Property Skewed by Ring Endomorphism
Chenar Abdul Kareem Ahmed, Renas Tahsin M. Salim · New Trends in Mathematical Science · 2021
The notion of an α-skew reflexive idempotent ring has been introduced in this paper to extend the concept of skew reflexive idempotent ring and that of an α-rigid ring.First basic properties of α-skew reflexive idempotent rings have been considered, including some examples needed in the process.It has been prove that for a ring R with an endomorphism α and n ≥ 2, if R satisfies the condition "eR f R f R = 0 implies eR f = 0 "and R is a right α-skew RIP ring, then V n (R) is a right ᾱ-skew RIP ring.Also it has proven that if R is an algebra over a field K and D the Dorroh extension of R by K, where α is an endomorphism of R with α(1) = 1, then R is a right α-skew RIP ring if and only if D is a right ᾱ-skew RIP ring.It's shown that if M is a multiplicative closed subset of a ring R consisting of central regular elements and α an automorphism of R, then R is right α-skew RIP if and only if M -1 R is right ᾱ-skew RIP.