Prereductions of ideals in local rings

Abstract

This paper contributes several results to the analytic theory of ideals. The main new concept is that of a prereduction (together with the closely related type-prereduction): if R is a ring and I (proper subset) are proper ideals in R, then A is called a prereduction of I in case A is not a reduction of I, but each ideal between A and I is a reduction of I. It is shown that each non-nilpotent proper ideal in a Noetherian ring has at least one prereduction, and a number of the basic properties of the ideals in the set of all prereductions of I are proved. Also, if I is a non-nilpotent proper ideal in a local (Noetherian) ring (R, M), then:There is a natural one-to-one correspondence between the set of the equivalence classes of the type-prereductions of I and the set of the maximal relevant ideals in the Rees ring of I (that is, the homogeneous prime ideals Q in the ring (t an indeterminate) such that and the Krull dimension of is equal to one). is the set of maximal elements in for all positive integers n, and, for each for all large integers m.A complete description is given of the prereductions and the elements in for each ideal I that is generated by analytically independent elements.If R/M is algebraically closed and I is a non-nilpotent and non-principal ideal in R, then there is a natural one-to-one correspondence between the sets and and for all positive integer n.