Abstract
The current paper is dedicated to the study of the classical K1 groups of graded rings. Let A be a Γ graded ring with identity 1, where the grading Γ is an abelian group. We associate a category with suspension to the Γ graded ring A. This allows us to construct the group valued functor K1 of graded rings. It will be denoted by K1gr. It is not only an abelian group but also a ℤ[Γ]-module. From the construction, it follows that there exists “locally” a matrix description of K1gr of graded rings. The matrix description makes it possible to compute K1gr of various types of graded rings. The K1gr 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 K1gr of cross products.
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