  The current paper is dedicated to the study of the classical $K_1$ groups of
graded rings. Let $A$ be a $\Gamma$ graded ring with identity $1$, where the
grading $\Gamma$ is an abelian group. We associate a category with suspension
to the $\Gamma$ graded ring $A$. This allows us to construct the group valued
functor $K_1$ of graded rings. It will be denoted by $K_1^{gr}$. It is not only
an abelian group but also a $\mathbb Z[\Gamma]$-module. From the construction,
it follows that there exists "locally" a matrix description of $K_1^{gr}$ of
graded rings. The matrix description makes it possible to compute $K_1^{gr}$ of
various types of graded rings. The $K_1^{gr}$ satisfies the well known
$K$-theory exact sequence $$ K_{1}^{gr}(A,I)\to K_1^{gr}(A)\to K_1^{gr}(A/I) $$
for any graded ideal $I$ of $A$. The above is used to compute $K^{gr}_1$ of
cross products.
