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• # Artykuł - szczegóły

## Colloquium Mathematicum

2007 | 109 | 1 | 1-7

## On the uniform behaviour of the Frobenius closures of ideals

EN

### Abstrakty

EN
Let 𝔞 be a proper ideal of a commutative Noetherian ring R of prime characteristic p and let Q(𝔞) be the smallest positive integer m such that $(𝔞^{F})^{[p^m]} = 𝔞^{[p^m]}$, where $𝔞^{F}$ is the Frobenius closure of 𝔞. This paper is concerned with the question whether the set ${Q(𝔞^{[p^m]}) : m ∈ ℕ₀}$ is bounded. We give an affirmative answer in the case that the ideal 𝔞 is generated by an u.s.d-sequence c₁,..., cₙ for R such that
(i) the map $R/∑_{j=1}^{n} Rc_j → R/∑_{j = 1}^{n} Rc²_j$ induced by multiplication by c₁...cₙ is an R-monomorphism;
(ii) for all $𝔭 ∈ ass(c₁^j, ..., cₙ^j)$, c₁/1,..., cₙ/1 is a $𝔭R_{𝔭}$-filter regular sequence for $R_{𝔭}$ for j ∈ {1,2}.

1-7

wydano
2007

### Twórcy

autor
• Department of Mathematics, Ferdowsi University of Mashhad, P.O. Box 1159-91775, Mashhad, Iran
• Institute for Studies, in Theoretical Physics and Mathematics, P.O. Box 19395-5746, Tehran, Iran