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\({\mathfrak m}\)-full ideals. (English) Zbl 0623.13012
From author’s introduction: An ideal \({\mathfrak a}\) of a local ring (R,\({\mathfrak m})\) is called \({\mathfrak m}\)-full if \({\mathfrak am}:y={\mathfrak a}\) for some y in a certain faithfully flat extension of R. The definition is due to D. Rees (unpublished) and he had obtained some elementary results (also unpublished). The present paper concerns some basic properties of \({\mathfrak m}\)-full ideals. One result is the characterization of \({\mathfrak m}\)-fullness in terms of the minimal number of generators of ideals, generalizing his result in low dimensional case (theorem 2, § 2). Meanwhile D. Rees asked me for which ideals is it true that \(\mu({\mathfrak a})\geq \mu ({\mathfrak b})\) for all \({\mathfrak b}\) containing \({\mathfrak a}\). Surprisingly enough it turns out \({\mathfrak m}\)-full ideals do have this property (theorem 3, § 2).
Reviewer: H.Yanagihara

MSC:
13H99 Local rings and semilocal rings
13A15 Ideals and multiplicative ideal theory in commutative rings
13E15 Commutative rings and modules of finite generation or presentation; number of generators
13B02 Extension theory of commutative rings
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References:
[1] Monog 16 (1972)
[2] Proceedings of Conference on Commutative Algebra and Combinatorics (1985)
[3] DOI: 10.1017/S0305004100043620 · doi:10.1017/S0305004100043620
[4] Commutative Algebra II, Van Nostrand (1960)
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