Primary ideals with finitely generated radical in a commutative ring. (English) Zbl 0794.13002

Gilmer, Robert; Heinzer, William

Manuscr. Math. 78, No. 2, 201-221 (1993).

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.

Reviewer: D.D.Anderson (Iowa City)

MSC:

13A15	Ideals and multiplicative ideal theory in commutative rings
13F20	Polynomial rings and ideals; rings of integer-valued polynomials
13E15	Commutative rings and modules of finite generation or presentation; number of generators

Keywords:

primary ideal; finitely generated prime ideal; polynomial ring over a primarily finite ring

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