Sums of automorphisms of free modules and completely decomposable groups

Abstract

For certain groups and modules we discuss the property that every endomorphism of that group or module is a sum of two automorphisms. Firstly, we consider the general case of a free R-module, M, of countably infinite rank where R is an associative unital ring such that for some positive integer m every free R-module of finite rank m has the above property. We prove that M also has the property and extend this result to free modules of uncountable rank. We deduce, when R is an elementary divisor ring, that every endomorphism of a free R-module of rank greater than 1 is a sum of two automorphisms. We apply these results to completely decomposable groups and make some interesting deductions about the expression of each endomorphism of such a group as a sum of a fixed number of automorphisms.