Generalized Prime Ideal Factorization of Submodules

In this article, we introduce generalized prime ideal factorization for all proper submodules of a finitely generated module over a Noetherian ring. We show that the generalized prime ideal factorization of a product of two coprime ideals is the product of the generalized prime ideal factorization of the ideals. We find conditions under which the generalized prime ideal factorization of a product of prime ideals is equal to the product of the prime ideals. We show that if R is a Dedekind domain, the generalized prime ideal factorization of an ideal I in R is exactly the prime ideal factorization of I.

Keywords:

Prime submodules , prime filtration , regular prime extension filtration