Modules over the algebra of the noncommutative equation yx = 1

Abstract

In this paper the associative algebra A generated by two elements x, y with defining relation y x = 1 is considered. An explicit description of the system of right ideals of A is obtained. Also a structure theorem for right modules over A in terms of extensions of modules is given. One can view A as the algebra of a non-commutative torus and one can construct a non-commutative projective line by adjoining points at infinity. Results about the Lie algebra of derivations of A and the group of automorphisms of A are derived.