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 presentation of $A_e$ in terms of generators and relations, which depends on the given presentation of $A$ and the choice of ``frames'', which identify each $A_g$ with a direct summand of a free $A_e$ module of finite rank. We use this result to describe the $0$-components of canonically $\mathbb{Z}$-graded Leavitt algebras and Leavitt path algebras of finite hypergraphs without sinks and sources.