On the expected number of generators of a submodule of a free module over a finite principal ideal ring.

摘要

This work concerns the expected minimal cardinality of a generating set of an arbitrary submodule of a finitely generated free module over a finite (commutative) principal ideal ring (PIR). Our primary objective is to obtain ring- and module-theoretic generalizations of the work that had been done earlier in the finite field-vector space situation by Dobbs and Lancaster.;In Chapter I, we provide background material, conventions, and some of the definitions needed for the work in Chapters II–IV.;Chapter II addresses finding the expected minimal cardinality of a generating set of an arbitrary submodule of a finitely generated free module over a finite special principal ideal ring (SPIR). We generalize the various “weighted” limiting expected values found by Dobbs in the field-vector space situation to an SPIR-module setting which Dobbs had explored in a special case. As a byproduct, we find a formula for the number of submodules of a given finitely generated free module (over a finite SPIR) which require a prescribed number of generators.;In Chapter III, using a basic structure theorem of PIRs and results from Chapter II, we obtain limiting expected values for the PIR-module situation. Here we investigate the various “weighting” contexts for limiting expected values which were introduced by Dobbs and we also address some other “weighting” contexts that do not arise naturally for vector spaces.;In contrast to the PIR settings in Chapters II and III, Chapter IV investigates some ring-module situations which have limiting expected values that behave in qualitatively different ways from the answers found in the (S)PIR settings.