On the embedded primary components of ideals. III (Q1345917)

Let \((R, {\mathfrak m})\) be a local Noetherian ring and \(I\) an ideal in \(R\) such that \({\mathfrak m}\) is an embedded prime of \(I\). In this third part of a series of four papers [part I: J. Algebra 167, No. 3, 724-744 (1994; see the preceding review), part II: J. Pure Appl. Algebra (to appear)] the authors give further results on the maximal \({\mathfrak m}\)-primary embedded components of \(I\) also called MEC's for brief. Generalizing a result from the first part it is shown that the number of ideals in an irredundant irreducible decomposition \({\mathfrak q} = \bigcap_ i {\mathfrak q}_ i\) of a MEC \({\mathfrak q}\) is an invariant of \(I\). The authors also investigate properties of the set \(IC ({\mathfrak q})\) of irreducible components of an \({\mathfrak m}\)-primary component of \(I\). E.g. they show that it is just the set of irreducible ideals \({\mathfrak Q}\) containing \({\mathfrak q}\) but not \({\mathfrak q} : {\mathfrak m}\). Moreover, \({\mathfrak q}\) is a MEC of \(I\) iff no ideal in \(IC ({\mathfrak q})\) contains \(I : {\mathfrak m}\). [For part IV of this paper see Trans. Am. Math. Soc. 347, No. 2, 701-708 (1995; see the following review].