Parametric decomposition of monomial ideals, II

Abstract

Letx1,...,xdbe anR-sequence in a commutative ringRand letIbe a monomial idea (soIis generated by elements of the formx1e1···xded, where eacheiis a nonnegative integer). The main results of this paper: (a) establish a practical formula which computes the monomial length ofIwhen Rad(I)=Rad((x1,...,xd)R); (b) determine necessary and sufficient conditions for the intersection of finitely many monomial ideals to again be a monomial ideal; (c) show that ifC, the set of all monomial ideals inRthat containI, is closed under finite intersections, then each idealJinChas a unique decomposition as an irredundant finite intersection of ideals of the form (xτ(1)a1,...,xτ(h)ah)R, where τ is a permutation of {1,...,d},h∈{1,...,d}, anda1,...,ahare positive integers; and, (d) give additional results for certain form rings and Rees rings ofR, related to the unique parametric decomposition theorem.

Department(s)

Mathematics

Document Type

Article

DOI

https://doi.org/10.1006/jabr.1997.6831

Publication Date

1-1-1997

Journal Title

Journal of Algebra

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