DELTA-REDUCTIONS OF MODULES

Let DELTA be a multiplicatively closed set of finitely generated nonze ro ideals of a ring R. Then the concept of a DELTA-reduction of an R - submodule D of an R -module A is introduced and several basic properti es of such reductions are established. Among these are that a minimal DELTA -reduction B of D exists and that every minimal basis of B can b e extended to a minimal basis of all R -submodules between B and D, wh en R is local and A is a finite R -module. Then, as an application, DE LTA -reductions B of a submodule C with property () are introduced, c haracterized, and shown to be quite plentiful. Here, () means that (R ,M) is a local ring of altitude at least one, that DELTA = {M(n); n gr eater-than-or-equal-to 0}, and that if D subset-or-equal-to E are R -s ubmodules between B and C, then every minimal basis of D can be extend ed to a minimal basis of E.