On the Embedded Primary Components of Ideals .1. Original Research Article

It is well known that an embedded primary component of an ideal I in a Noetherian ring R is not uniquely determined by I. Our main results are concerned with these embedded primary components of I. Specifically, they concern the maximal M-primary components of a non-open ideal I in a local ring (R, M). We show that if J is any ideal between I and a maximal M-primary component of I, then J is the intersection of the maximal M-primary components of I that contain J. Also, we characterize the sum of all the maximal M-primary components of I, show that one maximal M-primary component of I is irreducible if and only if all are, and then show that some other standard properties of M-primary ideals (length, number of generators, etc.) are not shared by different maximal M-primary components of I.