On the Embedded Primary Components of Ideals, III

Abstract

The results in this paper expand the fundamental decomposition theory of ideals pioneered by Emmy Noether. Specifically, let $ I$ be an ideal in a local ring $ (R,M)$ that has $ M$ as an embedded prime divisor, and for a prime divisor $ P$ of $ I$ let $ I{C_P}(I)$ be the set of irreducible components $ q$ of $ I$ that are $ P$-primary (so there exists a decomposition of $ I$ as an irredundant finite intersection of irreducible ideals that has $ q$ as a factor). Then the main results show: (a) $ I{C_M}(I) = \cup \{ I{C_M}(Q);Q\;{\text{is a }}\operatorname{MEC} {\text{ of }}I\} $ ($ Q$ is a MEC of $ I$ in case $ Q$ is maximal in the set of $ M$-primary components of $ I$); (b) if $ I = \cap \{ {q_i};i = 1, \ldots ,n\} $ is an irredundant irreducible decomposition of $ I$ such that $ {q_i}$ is $ M$-primary if and only if $ i = 1, \ldots ,k < n$, then $ \cap \{ {q_i};i = 1, \ldots ,k\} $ is an irredundant irreducible decomposition of a MEC of $ I$, and, conversely, if $ Q$ is a MEC of $ I$ and if $ \cap \{ {Q_j};j = 1, \ldots ,m\} $ (resp., $ \cap \{ {q_i};i = 1, \ldots ,n\} $) is an irredundant irreducible decomposition of $ Q$ (resp., $ I$) such that $ {q_1}, \ldots ,{q_k}$ are the $ M$-primary ideals in $ \{ {q_1}, \ldots ,{q_n}\} $, then $ m = k$ and $ ( \cap \{ {q_i};i = k + 1, \ldots ,n\} ) \cap ( \cap \{ {Q_j};j = 1, \ldots ,m\} )$ is an irredundant irreducible decomposition of $ I$; (c) $ I{C_M}(I) = \{ q,q\;{\text{is maximal in the set of ideals that contain }}I\;{\text{and do not contain }}I:M\} $; (d) if $ Q$ is a MEC of $ I$, then $ I{C_M}(Q) = \{ q;Q \subseteq q \in I{C_M}(I)\} $; (e) if $ J$ is an ideal that lies between $ I$ and an ideal $ Q \in I{C_M}(I)$, then $ J = \cap \{ q;J \subseteq q \in I{C_M}(I)\} $; and, (f) there are no containment relations among the ideals in $ \cup \{ I{C_P}(I)$; $ P$ is a prime divisor of $ I$}.

Department(s)

Mathematics

Document Type

Article

DOI

https://doi.org/10.1006/jabr.1995.1012

Publication Date

1995

Journal Title

Journal of Algebra

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