Parametric Decomposition of Monomial Ideals, II Original Research Article

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 set membership, variant {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.