The property of universality for some monoid algebras over non-commutative rings

We define on an arbitrary ring $A$ a family of mappings $(\sigma_{x,y})$ subscripted with elements of a multiplicative monoid $G$. The assigned properties allow to call these mappings derivations of the ring $A$. A monoid algebra of $G$ over $A$ is constructed explicitly, and the universality property of it is shown.