On z-Ideals and z ◦ -Ideals of Power Series Rings

Abstract

Let R be a commutative ring with identity and R[[x]] be the ring of formal power series with coefficients in R. In this article we consider sufficient conditions in order that P[[x]] is a minimal prime ideal of R[[x]] for every minimal prime ideal P of R and also every minimal prime ideal of R[[x]] has the form P[[x]] for some minimal prime ideal P of R. We show that a reduced ring R is a Noetherian ring if and only if every ideal of R[[x]] is nicely-contractible (we call an ideal I of R[[x]] a nicely-contractible ideal if (I ∩ R)[[x]] ⊆ I). We will trivially see that an ideal I of R[[x]] is a z-ideal if and only if we have I = (I, x) in which I is a z-ideal of R and also we show that whenever every minimal prime ideal of R[[x]] is nicely-contractible, then I[[x]] is a z ◦ -ideal of R[[x]] if and only if I is an ℵ0-z ◦ -ideal