TITLE:
Nilpotent Elements and Nil-Reflexive Property of Generalized Power Series Rings
AUTHORS:
Eltiyeb Ali
KEYWORDS:
Left APP-Ring, Generalized Power Series Reflexive Ring, Nil Generalized Power Series Reflexive Ring, S-Quasi Armendariz Ring, Semiprime Ring, Semicommutative Ring
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.12 No.11,
November
21,
2022
ABSTRACT: Let R be a ring and (S,≤) a strictly ordered monoid. In this paper, we deal with a new approach to reflexive property for rings by using nilpotent elements, in this direction we introduce the notions of generalized power series reflexive and nil generalized power series reflexive, respectively. We obtain various necessary or sufficient conditions for a ring to be generalized power series reflexive and nil generalized power series reflexive. Examples are given to show that, nil generalized power series reflexive need not be generalized power series reflexive and vice versa, and nil generalized power series reflexive but not semicommutative are presented. We proved that, if R is a left APP-ring, then R is generalized power series reflexive, and R is nil generalized power series reflexive if and only if R/I is nil generalized power series reflexive. Moreover, we investigate ring extensions which have roles in ring theory.