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On the integral closure of ideals. (English) Zbl 0902.13003
Let \(A\) be a Noetherian ring and \(I\) an ideal. The integral closure \(\overline{I}\) of \(I\) is defined to be \(\{x\in A\mid x^n+ a_{n-1} x^{n-1}+\cdots a_1 x+a_n= 0\) for some \(a_i\in I^i\}\). The ideal \(I\) is said to be integrally closed if \(\overline{I}= I\). The integral closure has many relevance. For example, when \(A\) is normal, the blowing-up \(\text{Proj} \bigoplus_{n\geq 0} I^n\) is normal if and only if there is a positive integer \(k\) such that \(\overline{I^{kn}}= (\overline{I^k})^n\) for all \(n>0\). In the article under review, the authors give a criterion for a certain class of \(I\) to be integrally closed by using “colon”. First they consider a generically complete intersection ideal in a Cohen-Macaulay ring. Let \(L= I:\sqrt{I}\). Then \(I\) is integrally closed if and only if \[ \sqrt{I}= IL:L^2. \tag{1} \] Next the authors consider a generically Gorenstein ideal. Instead of the Cohen-Macaulayness of \(A\), we assume that \(A_{\mathfrak p}\) is regular for all minimal primes \({\mathfrak p}\) of \(A/I\). Then \(I\) is integrally closed if and only if \(I\) satisfies (1). In the last part of the paper, the authors give a sufficient condition for a height 2 complete intersection ideal to satisfy \(\overline{I^n}= \overline{I}^n\) for all \(n>0\).

13B22 Integral closure of commutative rings and ideals
13C14 Cohen-Macaulay modules
13E05 Commutative Noetherian rings and modules
13A15 Ideals and multiplicative ideal theory in commutative rings
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