Mathematics – Commutative Algebra
Scientific paper
2009-06-05
Mathematics
Commutative Algebra
34 pages, LaTeX; major revisions and rewriting, including the *adding of Sections 1.1, 1.2, 7; Theorems 1.1, 1.3, 3.9, 4.3; an
Scientific paper
Given a direct sum G of cyclic groups, we find a sharp bound for the minimal number of proper subgroups whose union is G. This generalizes to sums of cyclic modules over more general rings, and we are able to solve this covering problem for all local, Artinian, or Dedekind rings (and even for monoids). This and related questions have been the subject of much past study, but only for finite groups. Our methods differ from those found in the literature, and make it possible to work in the general setting of "Chinese rings", which generalize the Chinese Remainder Theorem (and include the above cases). We also address a related problem: given a finite set M that is a direct sum of cyclic R-modules, and a distinguished element m in M, what is the smallest number of cosets of proper R-submodules whose union is M \ {m} ? When R is the ring of integers, this question is related to various conjectures in the literature.
No associations
LandOfFree
The covering problem for Chinese rings does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with The covering problem for Chinese rings, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The covering problem for Chinese rings will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-521671