Unit and idempotent additive maps over countable linear transformations

Abstract

Let V be a countably generated right vector space over a field F and a E End(V F ) be a shift operator. We show that there exist a unit u and an idempotent e in End(V F ) such that 1 - u, a - u are units in End(V F ) and 1 - e, a - e are idempotents in End(V F ) . We also obtain that if D is a division ring D % Z 2 , Z 3 and V D is a D -module, then for every alpha E End(V D ) there exists a unit u E End(V D ) such that 1 - u, alpha - u are units in End(V D ) .