Universal Property of Group Ring

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Theorem

Let $R$ be a commutative ring with unity.

Let $G$ be a group.

Let $R[G]$ be the corresponding group ring.

Let $S$ be a commutative ring with unity.

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

Let $\beta : G \to R^\times$ be a group homomorphism, where $R^\times$ is the multiplicative group of $R$.


Then there exists a unique ring homomorphism from $R[G]$ to $S$ which extends $\phi$ and $\beta$.


Proof