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 A M; (c) MB is the irredundant intersection of ยต B irreducible ideals; (d) ยต C โค ยต B for each cover C of B. Moreover, if B is an M-primary submodule of A, then B * = MB A M is the smallest basically full submodule of A containing B and B โ 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 โ 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 โ D โ C * . If G M denotes the form ring of R with respect to M and G + M its maximal homogeneous ideal, we prove that M n = M n * 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. ๏ฃฉ 2002 Elsevier Science (USA)