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]].