On Nilpotent Elements, Weak Symmetry and Related Properties of Skew Generalized Power Series Rings

The skew generalized power series ring R[[S,ω]] is a ring construction involving a ring R, a strictly ordered monoid (S,≤), and a monoid homomorphism ω:S→End(R). The ring R[[S,ω]] is a common generalization of ring extensions such as (skew) polynomial rings, (skew) Laurent polynomial rings, (skew) power series rings, (skew) Laurent series rings, (skew) Mal’cev–Neumann series rings, and (skew) monoid rings. In this paper, we study the nilpotent elements of skew generalized power series rings and the relationships between the properties of the rings R and R[[S,ω]] expressed in terms of annihilators, such as weak symmetry, weak zip, and the nil-Armendariz and McCoy properties. We obtain results on transferring the weak symmetry and weak zip properties between the rings R and R[[S,ω]], as well as sufficient and necessary conditions for a ring R to be (S,ω)-nil-Armendariz or linearly (S,ω)-McCoy.