a-Transforms of Local Rings and a Theorem on Multiplicities of Ideals

In two papers, (5) and (6), D. G. Northcott and the author considered the notion of the reductions of an ideal a of a Noether ring A . A reduction of a is an ideal b contained in a which satisfies a r +1 = a r b for all sufficiently large r . This notion was inspired by the following elementary property of a reduction. Suppose that A is a local ring with maximal ideal m , and that a is m -primary. It is well known (Samuel (10)) that the length of the ideal a n is, for large values of n equal to P a ( n ) where P a ( n ) is a polynomial in n whose degree d is equal to the dimension of A . If we write the coefficient of n d in P a ( n ) in the form e ( a )/ d !, e ( a ) is a positive integer termed the multiplicity of a . If now b is a reduction of a , then b is also m -primary, and e ( b ) = e ( a ).