Abstract: In the category of Monoid-graded rings, by means of the idempotent pro perties in e- component of M-graded rings, we show that three kinds of M-graded rings admit a graded ring decomposition into finite indecomposable graded rings and that this indecomposable decomposition is unique in a strong sense.

Key words: Monoid-graded ring, homogeneous idempotent, graded Block