Algebra Defined by Ring Homomorphism is Algebra

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Let $R$ be a commutative ring.

Let $(S, +, *)$ be a ring.

Let $f : R \to S$ be a ring homomorphism.

Let the image of $f$ be a subset of the center of $S$.

Let $(S_R, *)$ be the algebra defined by the ring homomorphism $f$.

Then $(S_R, *)$ is an algebra over $R$.


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