On endomorphism rings of free modules. (Q1415020)

Motivated by the classical result that the ring of \(n\times n\) matrices over a principal ideal domain \(R\) is itself a principal (left and right) ideal ring, the author establishes, under some extra hypothesis, an equivalent condition for the right ideals of the endomorphism ring \(\text{End}_R(F)\) of a free module \(F\) to be principal. Thus the main theorem states that if \(F\) is a free \(R\)-module in which every submodule is an endomorphic image, then a right ideal of \(\text{End}_R(F)\) is principal if and only if it is closed in the finite topology. The article ends by mentioning several cases when the previous hypothesis on \(F\) is satisfied, namely provided: (i) \(F\) is a free module that has a basis of cardinality at least as large as the cardinality of \(F\) itself, and, in particular, when \(F\) is a free module of infinite rank over a finite or countably infinite ring; (ii) \(R\) is semisimple and \(F\) is any free module; (iii) \(R\) is a principal ideal domain and \(F\) is any free module of finite rank, in this case recovering the initial classical property.