ON NIL SKEW GENERALIZED POWER SERIES REFLEXIVE RINGS

Abstract

Let $R$ be a ring and $(S, \leq)$ a strictly ordered monoid. In this paper, we deal with a new approaches to reflexive property for rings by using nilpotent elements. In this direction we introduce the notions of $(S, \omega)$-reflexive and $(S, \omega)$-$nil$-reflexive. Examples are given that, $(S, \omega)$-$nil$-reflexive is not $(S, \omega)$-reflexive. Under some suitable conditions, we proved that, if $R$ is a right $APP$-ring, then $R$ is $(S, \omega)$-reflexive and $R$ be a semiprime ring with the $ACC$ on left annihilator ideals, $(S, \leq)$ an $a.n.u.p.$-monoid, then $R$ is $(S, \omega)$-reflexive. Also, we proved that, $R$ is $(S, \omega)$-$nil$-reflexive if and only if $R/I$ is $(S, \overline{\omega})$-$nil$-reflexive, $R$ is $(S, \omega)$-$nil$-reflexive if and only if $T_{n}(R)$ is $(S, \omega)$-$nil$-reflexive and we will show that, if $R$ is a right Noetherian ring, then $R$ is $(S, \omega)$-$nil$-reflexive. Moreover, we investigate ring extensions which have roles in ring theory.