A restriction on left invertability in a ring in terms of its left modules

Let R be a ring with 1 and let M be a left unital R-module. Suppose r \in R and m \in M, m \neq 0, such that r \cdot m = 0. Prove that r does not have a left inverse in R.

We begin with a lemma.

Lemma: Let M be a unital left R-module. The for all r \in R, r \cdot 0 = 0. Proof: Let m \in M. Note that r \cdot 0 + r \cdot m = r \cdot (0+m) = r \cdot m. Thus r \cdot 0 = 0_M. \square

Suppose there exists s \in R such that sr = 1. Now m = 1 \cdot m = (sr) \cdot m = s \cdot (r \cdot m) = s \cdot 0 = 0, a contradiction; hence no such left inverse exists.

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