Balanced factorisations

Any rational number can be factored into a product of several rationals whose sum vanishes. This simple but nontrivial fact was suggested as a problem on a maths olympiad for high-school students. We completely solve similar questions in all finite fields and in some other rings, e.g., in the complex and real matrix algebras. Also, we state several open questions.Comment: 7 pages. A Russian version of this paper is at http://halgebra.math.msu.su/staff/klyachko/papers.ht

The number of non-solutions to an equation in a group and non-topologizable torsion-free groups

It is shown that, for any pair of cardinals with infinite sum, there exist a group and an equation over this group such that the first cardinal is the number of solutions to this equation and the second cardinal is the number of non-solutions to this equation. A countable torsion-free non-topologizable group is constructed.Comment: 5 pages; minor changes in the introduction and reference