Vol. 15, No. 2, 1965

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Some containment relations between classes of ideals of a commutative ring

Robert William Gilmer, Jr.

Vol. 15 (1965), No. 2, 497–502

The first section of this paper is devoted to proving the following theorem. Let D be an integral domain with identity. Let 𝒫 be the set of prime powers of D, 𝒱 the set of valuation ideals of D, and let k be the quotient field of D. 𝒱𝒫 if and only if the following conditions hold: (i) Each prime ideal P of D defines a P-adic valuation in the sense of van der Waerden, and (ii) every valuation of k finite on D is isomorphic to a P-adic valuation for some P.

The second section considers three additional sets of ideals; the set 𝒬 of primary ideals, the set 𝒮 of semi-primary ideals, and the set 𝒜 of ideals A such that the complement of some prime ideal is prime to A.

Mathematical Subject Classification
Primary: 13.20
Secondary: 16.00
Received: 14 April 1964
Published: 1 June 1965
Robert William Gilmer, Jr.