Khare, AP and Tikaradze, A (2021) Covering modules by proper submodules. In: Communications in Algebra .
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Abstract
A classical problem in the literature seeks the minimal number of proper subgroups whose union is a given finite group. A different question, with applications to error-correcting codes and graph colorings, involves covering vector spaces over finite fields by (minimally many) proper subspaces. In this note we cover R-modules by proper submodules for commutative rings R, thereby subsuming and recovering both cases above. Specifically, we study the smallest cardinal number (Formula presented.) possibly infinite, such that a given R-module is a union of (Formula presented.) -many proper submodules. (1) We completely characterize when (Formula presented.) is a finite cardinal; this parallels for modules a 1954 result of Neumann. (2) We also compute the covering (cardinal) numbers of finitely generated modules over quasi-local rings and PIDs, recovering past results for vector spaces and abelian groups respectively. (3) As a variant, we compute the covering number of an arbitrary direct sum of cyclic monoids. Our proofs are self-contained. © 2021 Taylor & Francis Group, LLC.
| Item Type: | Journal Article |
|---|---|
| Publication: | Communications in Algebra |
| Publisher: | Taylor and Francis Ltd. |
| Additional Information: | The copyright for this article belongs to Authors |
| Department/Centre: | Division of Physical & Mathematical Sciences > Mathematics |
| Date Deposited: | 28 Nov 2021 09:06 |
| Last Modified: | 28 Nov 2021 09:06 |
| URI: | http://eprints.iisc.ac.in/id/eprint/69956 |
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