Universal Property of Group Ring
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Theorem
Let $R$ be a commutative ring with unity.
Let $G$ be a group.
Let $R \sqbrk G$ 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 $R \sqbrk G$ to $S$ which extends $\phi$ and $\beta$.
Proof
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