Primary ideals with finitely generated radical in a commutative ring

Abstract

Suppose p is a finitely generated prime ideal in a commutative ring R and q is a p-primary ideal. We show that in general q need not be finitely generated. We consider the class of primarily finite rings -- that is, rings R such that if p is a finitely generated prime ideal in R, then any p-primary ideal of R is finitely generated. If R is a polynomial ring in an arbitrary set of indeterminates over a Noetherian ring, we show that R is primarily finite and that an ideal c of R is finitely generated if and only if c has only finitely many associated prime ideals and each of the associated prime ideals of c is finitely generated. If X is a set of indeterminates over a ring R and p is a finitely generated prime ideal of R[X] that lies over a maximal ideal m of R, we show that m is finitely generated and p is primarily finite. We show that a polynomial ring in one variable over a Priifer domain is primarily finite, and present several results related to the question of whether a polynomial ring in several variables over a Priifer domain is primarily finite.