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

Let r be a commutative ring with identity and r[[x]] bethe ring of formal power series with coefficients in r. in this articlewe consider sufficient conditions in order that p[[x]] is a minimal primeideal of r[[x]] for every minimal prime ideal p of r and also everyminimal prime ideal of r[[x]] has the form p[[x]] for some minimalprime ideal p of r. we show that a reduced ring r is a noetherianring if and only if every ideal of r[[x]] is nicely-contractible (we call anideal i of r[[x]] a nicely-contractible ideal if (i r)[[x]]  i). we willtrivially see that an ideal i of r[[x]] is a z-ideal if and only if we havei = (i, x) in which i is a z-ideal of r and also we show that wheneverevery minimal prime ideal of r[[x]] is nicely-contractible, then i[[x]] isa z-ideal of r[[x]] if and only if i is an @0-z-ideal.