(Solved): 6. Let R be a ring and let FR be a subring. Suppose that F is a field. (Wh ...

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6. Let $$R$$ be a ring and let $$\mathbb{F}?R$$ be a subring. Suppose that $$\mathbb{F}$$ is a field. (When this happens, we say that $$R$$ is an $$\mathbb{F}$$-algebra.) Define scalar multiplication $$\mathbb{F}\times R?R$$ by $$c?r:=cr$$ (where the right-hand side denotes ring multiplication). Prove that this scalar multiplication and the usual ring addition turns $$R$$ into a vector space over $$\mathbb{F}$$.