On the embedded primary components of ideals. IV

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 $I$}.