Primary ideals with finitely generated radical in a commutative ring
This paper consider the question of when a finitely generated prime ideal \(P\) has every \(P\)-primary ideal finitely generated. The authors call a prime ideal with this property primarily finite and call a ring primarily finite if each of its finitely generated prime ideals is primarily finite. An example is given of a strongly Laskerian ring with only two prime ideals \(P \subset M\) and with \(P\) finitely generated and \(P\) having only one primary ideal different from \(P\) and with that primary ideal not finitely generated. Thus a finitely generated prime ideal need not be primarily finite. It is shown that while a polynomial ring over a primarily finite ring need not be primarily finite, a polynomial ring in one indeterminate over a Prüfer domain or in any number of indeterminates over a one-dimensional Prüfer domain or any Noetherian ring is primarily finite. The paper contains a number of other interesting related results.
polynomial ring over a primarily finite ring
scientific article; zbMATH DE number 492772