Basically Full Ideals in Local Rings

Abstract

Let A be a finitely generated module over a (Noetherian) local ring (R, M). We say that a nonzero submodule B of A is basically full in A if no minimal basis for B can be extended to a minimal basis of any submodule of A properly containing B. We prove that a basically full submodule of A is M-primary, and that the following properties of a nonzero M-primary submodule B of A are equivalent: (a) B is basically full in A; (b) B = (MB):AM; (c) MB is the irredundant intersection of μ(B) irreducible ideals; (d) μ(C) ≤ μ(B) for each cover C of B. Moreover, if B is an M-primary submodule of A, then B* := (MB):AM is the smallest basically full submodule of A containing B and B ↦ B* is a semiprime operation on the set of nonzero M-primary submodules B of A. We prove that all nonzero M-primary ideals are closed with respect to this operation if and only if M is principal. In relation to the closure operation B ↦ B*, we define and study the bf-reductions of an M-primary submodule D of A; that is, the M-primary submodules C of D such that C ⊆ D ⊆ C*. If G(M) denotes the form ring of R with respect to M and G+(M) its maximal homogeneous ideal, we prove that Mn = (Mn)* for all (resp. for all large) positive integers n if and only if grade(G+(M)) > 0 (resp. grade(M) > 0). For a regular local ring (R, M), we consider the M-primary monomial ideals with respect to a fixed regular system of parameters and determine necessary and sufficient conditions for such an ideal to be basically full.