2002/09/22 by Les Reid, Reid, Les, Leslie G. Roberts +3 · 1 citation
Mathematics · #Advanced Topology and Set Theory #Commutative Algebra and Its Applications #Rings, Modules, and Algebras #math.AC #math.AG #msc:13A20 #msc:13C13
paper · pdf · doi:10.48550/arxiv.math/0209285
19 pages
arxiv created 2002/09/22 · arxiv updated 2009/11/30
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 generated by all homogeneous elements of degree at least m and monomial ideals in a polynomial ring over a field. For ideals of the first trype we generalize a recent result of S. Faridi. We prove that a monomial ideal in a polynomial ring in n indeterminates 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 obtained by taking integral closures of m-primary ideals generated by powers of the variables. We obtain classes of normal monomial ideals and arithmetic critera for deciding when the monomial ideal is not normal.