On the formal power series algebras generated by a vector space and a linear functional

Let R be a vector space ( on C) and ϕ be an element of R lowast; (the dual space of R), the product r middot; s = ϕ(r)s converts R into an associative algebra that we denote it by Rϕ. We characterize the nilpotent, idempotent and the left and right zero divisor elements of Rϕ[[x]]. Also we show that the set of all nilpotent elements and also the set of all left zero divisor elements of Rϕ[[x]] are ideals of Rϕ[[x]].