Title

Some Results on Normal Homogeneous Ideals

Abstract

In this article we investigate when a homogeneous ideal in a graded ring is normal, that is, when all positive powers of the ideal are integrally closed. We are particularly interested in homogeneous ideals in an ℕ-graded ring Aof the form A ≥m ≔ ⊕ℓ≥m A ℓand monomial ideals in a polynomial ring over a field. For ideals of the form A ≥m we generalize a recent result of Faridi. We prove that a monomial ideal in a polynomial ring in nindeterminates over a field is normal if and only if the first n − 1 positive powers of the ideal are integrally closed. We then specialize to the case of ideals of the form I( λ ) ≔ , where J( λ ) = (,…, ) ⊆ K[x 1,…, x n ]. To state our main result in this setting, we let ℓ = lcm(λ1,…, ,…λ n ), for 1 ≤ i ≤ n, and set λ ′ = (λ1,…, λ i−1, λ i + ℓ, λ i+1,…, λ n ). We prove that if I( λ ′) is normal then I( λ ) is normal and that the converse holds with a small additional assumption.

Department(s)

Mathematics

Document Type

Article

DOI

https://doi.org/10.1081/agb-120022805

Keywords

normal monomial ideals, Ideals, integral closure

Publication Date

2003

Recommended Citation

Reid, Les, Leslie G. Roberts, and Marie A. Vitulli. "Some results on normal homogeneous ideals." Communications in Algebra 31, no. 9 (2003): 4485-4506.

Journal Title

Communications in Algebra

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