Over a ring with a central idempotent, every module is an internal direct sum

Let R be a ring with 1 and let M be a left unital R-module. Suppose there exists e \in R such that e^2 = e and e \in Z(R). Prove that M = eM \oplus (1-e)M.


First, recall by this previous exercise that eM and (1-e)M are both submodules of M. Suppose m \in M. Then m = e \cdot m + 1 \cdot m - e \cdot m \in eM + (1-e)M, so that M = eM + (1-e)M. Finally, suppose m \in eM \cap (1-e)M; say m = e \cdot a = (1-e) \cdot b. Now e \cdot a = 1 \cdot b - e \cdot b, so that e \cdot (a+b) = b. Now e \cdot b = e^2 \cdot (a+b) = e \cdot (a + b) = e \cdot a + e \cdot b, so that e \cdot a = 0. Thus m = 0, and we have eM \cap (1-e)M = 0. So M = eM \oplus (1-e)M.

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