Universal Property of Polynomial Ring

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Let $R$ be a commutative ring with unity.

The different definitions of a polynomial ring $\struct {\map R x, \iota, x}$ on $R$ satisfy the universal property of a polynomial ring:

Construction using Free Monoid on Set

Let $R, S$ be commutative and unitary rings.

Let $\family {s_j}_{j \mathop \in J}$ be an indexed family of elements of $S$.

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

Let $R \sqbrk {\set {X_j: j \in J} }$ be a polynomial ring.

Then there exists a unique evaluation homomorphism $\phi: R \sqbrk {\set {X_j: j \in J} } \to S$ at $\family {s_j}_{j \mathop \in J}$ extending $\psi$.

Also see