2018/11/16 by Adeleh Azari, Azari, Adeleh, Simin Mollamahmoudi +3
Computer Science · Mathematics · #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.1811.06881
openalex publication_date 2018/11/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let R be a commutative Noetherian ring and let bf x :=x1,\…,xd\nbe a regular R-sequence contained in the Jacobson radical of R. An ideal\nI of R is said to be a monomial ideal with respect to bf x if it is\ngenerated by a set of monomials x1e1\… xded. It is shown that,\nif bf xR is a prime ideal of R, then each monomial ideal I has a\ncanonical and unique decomposition as an irredundant finite intersection of\nprimary ideals of the form xe1\τ(1)R+\…+xes\τ(s)R, where\n\τ is a permutation of 1,\…,d , s\∈ 1,\…,d and\ne1,\…,es are the positive integers. This generalizes and provides a\nshort proof of the main results of citeHMRS, HRS. Also, we show that for\nevery integer k\≥1, I(k)=Ik, if and only if AssR R/Ik \⊆\n AssR R/I, whenever I is a squarefree monomial ideal, where I(k) is\nthe kth symbolic power of I. Moreover, in this circumstance it is shown\nthat all powers of I are integrally closed.\n