A Note on Skew Generalized Power Serieswise Reversible Property

Abstract

The aim of this paper is to introduce and study (S, ω)-nil-reversible rings wherein we call a ring R is (S, ω)-nil-reversible if the left and right annihilators of every nilpotent element of R are equal. The researcher obtains various necessary or sufficient conditions for (S, ω)-nil-reversible rings are abelian, 2-primal, (S, ω)-nil-semicommutative and (S, ω)-nil-Armendariz. Also, he proved that, if R is completely (S, ω)-compatible (S, ω)-nil-reversible and J an ideal consisting of nilpotent elements of bounded index ≤ n in R, then R/J is (S, ¯ω)-nil-reversible. Moreover, other standard rings-theoretic properties are given.

Some Results of Malcev-Neumann Rings

“…A ring is called APP if it is right APP and left APP. By Proposition 2.3 [35], the class of right APP-rings includes both left PP-rings and right p.q.-Baer rings (and hence it includes all biregular rings and all quasi-Baer rings), for some details to use this conditions see [11] and [17]. In [20] the authors showed that left p.q.-Baer rings are also right APP and provided various examples of commutative APP-rings which are neither p.q.-Baer nor PP.…”

The Reflexive Condition on Skew Monoid Rings

“…The investigation of the composition of the collection of nilpotent elements in noncommutative ring constructions is a crucial and highly active field in noncommutative algebra. This is evidenced by numerous studies conducted by various authors see [3], [1], [22], [15], [13], [4], [5], [2] and [18].…”