Monoid algebras over non-commutative rings

Abstract

We define on an arbitrary ring A a family of mappings (σx,y) subscripted with elements of a multiplicative monoid G. The assigned properties allow to call these mappings as derivations of the ring A. Beside the general situation it is given their description for the case of a partially ordered monoid. A monoid algebra of G over A is constructed explicitly, and the universality property of it is shown. The notion of a monoid algebra in our context extends those of a group ring, a skew polynomial ring, Weyl algebra and other related ones. The connection with crossed products is also shown.