Associated Primes and Primary Decomposition

Abstract

As we have suggested in Chapter 1, the earliest impulse toward the development of what is now commutative algebra came from the desire of the number theorists to make use of unique factorization in rings of integers in number fields other than Q. When it became clear that unique factorization did not always hold, the search for the strongest available alternative began. The theory of primary decomposition is the direct result of that search. Given an ideal I in a Noetherian ring R, the theory identifies a finite set of “associated” prime ideals of R, and tells how to “decompose” I as an intersection of “primary” ideals that are closely connected with these prime ideals. More generally, the theory produces such a set of associated primes and a decomposition of any submodule of a finitely generated R-module.