Skew group ring

Summary

The skew group ring $R * G$ of a group $G$ acting on a ring $R$ by automorphisms is like the group ring $R[G]$. The difference is that group elements and coefficients do not commute: $g r = g(r) g$. It is the algebraic counterpart of the C*-algebra crossed product. It turns a group action on a ring into a single ring, and its modules are the $R$-modules with a compatible action of $G$. It is used in invariant theory, Galois theory of rings and the representation theory of finite groups.