1974
DOI: 10.2140/pjm.1974.52.185
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Locally quasi-unmixed Noetherian rings and ideals of the principal class

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Cited by 52 publications
(21 citation statements)
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References 12 publications
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“…Using this last result, McAdam proved, in [3,Theorem 3], the very interesting result that if / C P are ideals in a Noetherian domain R that satisfies the altitude formula and if P is prime, then P is a prime divisor of the integral closures of all large powers of /" if and only if (height P) equals the analytic spread of IRP. Our main results, Theorems 1 and 2, were suggested by McAdam's result and the related result [6,Theorem 2.29], and they characterize Noetherian domains that satisfy the altitude formula in terms of these asymptotic prime divisors and analytic spreads. Then, after proving several corollaries of these theorems, we close by showing certain additional rings (besides those in [3,Theorem 6]) have the property that a prime ideal is a prime divisor of all large powers of an ideal / if and only if it is a prime divisor of the integral closures of all large powers of /.…”
mentioning
confidence: 84%
“…Using this last result, McAdam proved, in [3,Theorem 3], the very interesting result that if / C P are ideals in a Noetherian domain R that satisfies the altitude formula and if P is prime, then P is a prime divisor of the integral closures of all large powers of /" if and only if (height P) equals the analytic spread of IRP. Our main results, Theorems 1 and 2, were suggested by McAdam's result and the related result [6,Theorem 2.29], and they characterize Noetherian domains that satisfy the altitude formula in terms of these asymptotic prime divisors and analytic spreads. Then, after proving several corollaries of these theorems, we close by showing certain additional rings (besides those in [3,Theorem 6]) have the property that a prime ideal is a prime divisor of all large powers of an ideal / if and only if it is a prime divisor of the integral closures of all large powers of /.…”
mentioning
confidence: 84%
“…It follows that, for all positive integers n, u n R = {u n R q ∩ R | q ∈ Ass(R/(uR))} (by [9, (12.6)]) and that each u n R q ∩ R is integrally closed, so u n R = (u n R) a , by [11,Lemma 4]. Therefore I n = u n R ∩ R = (u n R) a ∩ R = I n a (by [12,Lemma 2.5]) for all positive integers n, so it follows that I is a normal ideal. 2…”
Section: Examples Of Ideals With Some Rees Integer Equal To Onementioning
confidence: 96%
“…p is a regular local ring, p h R ′ p is integrally closed, and as R ′ is finitely generated over a locally formally equidimensional (regular) ring, R ′ q is locally formally equidimensional for every prime ideal q containing p. By a theorem of Ratliff, from [14], since p is generated by a regular sequence, the integral closure of p h R ′ q has no embedded prime ideals. It follows that the integral closure of…”
Section: Integral Closures Of (General) Monomial Idealsmentioning
confidence: 99%