Algebraic Aspects of Commutation of Linear Operators up to a Factor

Abstract

When representing projective geometry by means of a vector space, commutativity can be replaced by commutativity up to a factor. This feature was investigated by F. Cecioni under very weak assumptions, but it is hard to generalize the methods of [4] to a wider algebraic context. In this note, we develop the independent treatment of H. Weyl, and extend the approach of to non-commutative rings under suitable assumptions on the endomorphisms. From this point of view, we show that commutativity of operators up to a non-trivial factor is an exceptional phenomenon in comparison to strict commutativity.