Generalized Baеr and Generalized Quasi-Baеr Properties of Skеw Generalized Power Series Rings

Let R be a ring with identity, (S,≤) an ordered monoid, ω:S→End(R) a monoid homomorphism, and A=R[[S,ω]] the ring of skew generalized power series. The concepts of generalized Baer and generalized quasi-Baer rings are generalization of Baer and quasi-Baer rings, respectively. A ring R is called generalized right Baer (generalized right quasi-Baer) if for any non-empty subset S (right ideal I) of R, the right annihilator of 〖S 〗^n (I^n) is generated by an idempotent for some positive integer n. Left cases may be defined analogously. A ring R is called generalized Baer (generalized quasi-Baer) if it is both generalized right and left Baer (generalized right and left quasi-Baer) ring. In this paper, we examine the behavior of a skew generalized power series ring over a generalized right Baer (generalized right quasi-Baer) ring and prove that, under specific conditions, the ring A is generalized right Baer (generalized right quasi-Baer) if and only if R is a generalized right Baer (generalized right quasi-Baer) ring.