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Let 
be a ring graded by an arbitrary grading abelian group Γ. We say that R is a uniformly graded-coherent ring if there is a map ϕ : ℕ → ℕ such that for every n ∈ ℕ, and any nonzero graded R-module homomorphism 
of degree 0, where λ1 , … ,λn are degrees in Γ, ker f can be generated by ϕ(n) homogeneous elements. In this paper, we provide the elementary properties of uniformly graded-coherent rings.
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