Dimension theory in power series rings

each ie ω0} and we define A R [[X]] to be the ideal of R [[X]] which is generated by A. Then A R [[X]] = {f(X): Af S B for some finitely generated ideal B of R with B £ A}. It is clear that A iϋ [[X]] S A [[X]]; equality holds if and only if each countably generated ideal of R contained in A is contained in a finitely generated ideal of R contained in A. In particular, if V is a valuation ring containing an ideal A which is countably generated but not finitely generated, then A V[[X]] c A [[X]]. Finally, we note that if A is an ideal of i2, then R [[X]]/A [[X]] ~ (R/A) [[X]]; hence A [[X]] is a prime ideal of R [[X]] if and only if A is a prime ideal of R.