Regular divisors of a submodule

In this article, we extend the concept of divisors to ideals of Noetherian rings, more generally, to submodules of finitely generated modules over Noetherian rings. For a submodule N of a finitely generated module M over a Noetherian ring, we say a submodule K of M is a regular divisor of N in M if K occurs in a regular prime extension filtration of M over N. We show that a submodule N of M has only a finite number of regular divisors in M. We also show that an ideal \mathfrak b is a regular divisor of a non-zero ideal \mathfrak a in a Dedekind domain R if and only if \mathfrak b contains \mathfrak a. We characterize regular divisors using some ordered sequences of prime ideals and study their various properties. Lastly, we formulate a method to compute the number of regular divisors of a submodule by solving a combinatorics problem.

Keywords:

Maximal independent subset , Partially ordered set , Prime ideal factorization , Regular divisors , Regular prime extension filtration