Ideals of finite codimension in free algebras and the \(fc\)-localization
The topology defined by all finite codimensional right ideals of an algebra over a field is an analogue of the profinite topology of group theory. In the case of the free algebra and the group ring of a free group it has many interesting properties. For example, we show that its open ideals are precisely the finitely generated essential ones. Also, finitely generated right ideals are closed. This topology is what is called a ``Gabriel'' topology, and following Gabriel's general procedure it gives rise to a localization of the algebra which we call the \(fc\)- localization. In the free case this localization does not have the unique rank property and we show that its ``Leavitt'' numbers are 1 and \(n-1\). This means that two free modules over it are isomorphic if and only if their ranks differ by a multiple of \(n-1\). The proofs are, for the most part, applications of varying complexity of Schreier's method.
scientific article; zbMATH DE number 537389