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Full ideals. (English) Zbl 1171.13012

Summary: Contractedness of \(\mathfrak m\)-primary integrally closed ideals played a central role in the development of Zariski’s theory of integrally closed ideals in two-dimensional regular local rings (\(R, \mathfrak m\)). In such rings, the contracted \(\mathfrak m\)-primary ideals are known to be characterized by the property that \(I: \mathfrak m = I: x\) for some \(x \in \mathfrak m \setminus \mathfrak m^{2}\). We call the ideals with this property full ideals and compare this class of ideals with the classes of \(\mathfrak m\)-full ideals, basically full ideals, and contracted ideals in higher dimensional regular local rings. The \(\mathfrak m\)-full ideals are easily seen to be full. In this article, we find a sufficient condition for a full ideal to be \(\mathfrak m\)-full. We also show the equivalence of the properties full, \(\mathfrak m\)-full, contracted, integrally closed, and normal, for the class of parameter ideals. We then find a sufficient condition for a basically full parameter ideal to be full.

MSC:

13H05 Regular local rings
13H15 Multiplicity theory and related topics
13B22 Integral closure of commutative rings and ideals
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References:

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