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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 $$n>0$$.

##### MSC:
 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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