On the embedded primary components of ideals. I (Q1333215)

This is the first in an interesting series of four papers where the authors study properties of embedded primary components of ideals. Let \((R, {\mathfrak m})\) be a local Noetherian ring and \(I\) an ideal in \(R\) such that \({\mathfrak m}\) is an embedded prime of \(I\). Then it is shown that every \({\mathfrak m}\)-primary component, say \({\mathfrak q}\), of \(I\) is contained in a maximal \({\mathfrak m}\)-primary component of \(I\) and that \({\mathfrak q}\) is the intersection of all such maximal \({\mathfrak m}\)-primary components. Moreover \(I\) is the intersection of all maximal \({\mathfrak m}\)-primary components of \(I\). This result is the starting point for the authors to investigate systematically properties of such maximal embedded components of \(I\) which are called MEC's for brief. They give a characterization of the union as well of the sum of all MEC's. Finally they prove that one MEC of \(I\) is an irreducible ideal iff this holds for all MEC's, and they give examples that other standard properties as length, minimal number of generators, being integrally closed, are not shared by different MEC's of \(I\). [For part II--IV see J. Pure Appl. Algebra (to appear), J. Algebra 171, No. 1, 272-293 (1995) and Trans. Am. Math. Soc. 347, No. 2, 701-708 (1995; see the two following reviews)].