On z-Ideals and z^0-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 @0-z-ideal.