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 problem generalizes to sums of cyclic modules over more general rings, such as local and Artinian rings or Dedekind domains and reduces to covering vector spaces by proper subspaces. As a consequence, we are also able to solve the analogous problem for monoids.

If M is a simple module over a ring R then, by the Schur’s lemma, the endomorphism ring of M is a division ring. However, the converse of this result does not hold in general, even when R is artinian. In this short note, we consider perfect rings for which the converse assertion is true, and we show that these rings are exactly the primary decomposable ones.

We consider the following condition (*) on an associative ring R : (*). There exists a function f from R into R such that f is a group homomorphism of (R,+), f is injective on R2, and f(xy) = (xy)n(x,y) for some positive integer n(x,y) > 1. Commutativity and structure are established for Artinian rings R satisfying (*), and a counterexample is given for nonArtinian rings. The results generalize...