An Alternative Perspective on Skew Generalized Power Series Rings

This paper continues the ongoing effort to study the structure of the set of nilpotent elements in noncommutative ring constructions. Let R be any ring, $${(S,\leq)}$$ a strictly (partially) ordered monoid and also $${\omega:S\rightarrow}$$ End(R) a monoid homomorphism. A skew generalized power series ring $${R[[{S,\omega\leq}}]]$$ consists of all functions from a monoid S to a coefficient ring R, whose support contains neither infinite descending chains nor infinite antichains, with pointwise addition, and with multiplication given by convolution twisted by an action $${\omega}$$ of the monoid S on the ring R. Our studies in this paper is strongly connected to the question of whether or not a skew generalized power series ring $${R[[{S,\omega\leq}}]]$$ over a nil coefficient ring R is nil, which is related to the famous question of Amitsur. However, we show that under mild “Armendariz-like” hypothesis on a coefficient ring R, we obtain stronger conditions on the coefficients of elements of a skew generalized power series ring $${R[[{S,\omega\leq}}]]$$ . We will also explore some annihilator conditions in the skew generalized power series ring setting, unifying and generalizing a number of known Armendariz-like and McCoy-like conditions in the special cases.