ON THE EMBEDDED PRIMARY COMPONENTS OF IDEALS .4.

The results in this paper expand the fundamental decomposition theory of ideals pioneered by Emmy Noether. Specifically, let I be an ideal i n a local ring (R, M) that has M as an-embedded prime divisor, and for a prime divisor P of I let ICp(I) be the set of irreducible component s a of I that are P-primary (so there exists a decomposition of I as a n irredundant finite intersection of irreducible ideals that has q as a factor). Then the main results show: (a) ICM(I) = boolean OR{ICM(Q); Q is a MEC of I} (Q is a MEC of I in case and is maximal in the set of M-primary components of I); (b) if I = boolean AND{q(i);i = 1,..., n) is an irredundant irreducible decomposition of I such that q(i) is M- primary if and only if i = 1,..., k < n, then boolean AND{q(i);i = 1,. .., k} is an irredundant irreducible decomposition of a MEC of I, and, conversely, if Q is a MEC of I and if boolean AND{Q(i);j = 1,...,m} ( resp., n(qi; i = 1,..., n)) is an irredundant irreducible decompositio n of Q (resp., I) such that q(l),...,q(k) are the M-primary ideals in {q(l),...,q(n)}, then m = k and (boolean AND{q(i);i = k+1,..., n}) boo lean AND (boolean AND{Q(j);j = 1,..., m}) is an irredundant irreducibl e decomposition of I; (c) ICM(I) = (q; a is maximal in the set of idea ls that contain I and do not contain I: M); (d) if Q is a MEC of I, th en ICM(Q) = (q; Q subset of or equal to q is an element of ICM(I)}; (e ) if J is an ideal that lies between I and an ideal Q is an element of ICM(I) then J = boolean AND{q;J subset of or equal to q is an element of ICM(I)}; and, (f) there are no containment relations among the ide als in boolean OR{ICp(I); P is a prime divisor of I}.