On the identity component of a strongly graded ring - with applications to graph algebras

Let $G$ be a group, $K$ a commutative ring and $A$ a strongly $G$-graded $K$-algebra with a given homogeneous presentation. We show that the identity component $A_e$ can be obtained by successively applying some of Bergman's universal ring constructions to the underlying ring $K$. This yields a pres