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Let S be a semigroup and let be an S-graded ring. Rs = 0 for all but finitely many elements s ∈ S1, then R is said to have finite support. In this paper we concern ourselves with the question of whether a graded ring R with finite support inherits a given ring theoretic property from the homogeneous subrings Re corresponding to idempotent semigroup elements e.

Necessary and sufficient conditions are given for a generalised matrix ring to be semiprime right Goldie.

Let S be a semigroup with no infinite subgroups and let R be a right Artinian S-graded ring. We prove that R necessarily has finite support.

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