Universal Property of Polynomial Ring/Free Monoid on Set

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Theorem

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

Let $\left\langle{s_j}\right\rangle_{j \mathop \in J}$ be an indexed family of elements of $S$.

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

Let $R \left[{\left\{{X_j: j \in J}\right\}}\right]$ be a polynomial ring.


Then there exists a unique evaluation homomorphism $\phi: R \left[{\left\{{X_j: j \in J}\right\}}\right] \to S$ at $\langle s_j\rangle_{j \in J}$ extending $\psi$.


Proof

Let $Z$ be the set of all multiindices indexed by $J$.

Let $k_j$ be the $j$th component of a multiindex $k$.

Let $\displaystyle f = \sum_{k \mathop \in Z} a_k \prod_{j \mathop \in J} X_j^{k_j}$ be a polynomial over $R$.


Define:

$\displaystyle \phi \left({f}\right) = \sum_{k \mathop \in Z} \psi \left({a_k}\right) \prod_{j \mathop \in J}s_j^{k_j}$

It is clear that $\phi$ extends $\psi$.

If $\displaystyle g = \sum_{k \mathop \in Z} b_k \prod_{j \mathop \in J} X_j^{k_j}$, then:

\(\ds \phi \left({f + g}\right)\) \(=\) \(\ds \phi \left({\sum_{k \mathop \in Z} \left({a_k + b_k}\right) \prod_{j \mathop \in J} X_j^{k_j} }\right)\) Definition of Addition of Polynomial Forms
\(\ds \) \(=\) \(\ds \sum_{k \mathop \in Z} \left({\psi \left({a_k + b_k}\right)}\right) \prod_{j \mathop \in J} s_j^{k_j}\) Definition of $\phi$
\(\ds \) \(=\) \(\ds \sum_{k \mathop \in Z} \left({\psi \left({a_k}\right) + \psi \left({b_k}\right)}\right) \prod_{j \mathop \in J} s_j^{k_j}\) Definition of Ring Homomorphism
\(\ds \) \(=\) \(\ds \sum_{k \mathop \in Z} \psi \left({a_k}\right) \prod_{j \mathop \in J} s_j^{k_j} + \sum_{k \mathop \in Z} \psi \left({b_k}\right) \prod_{j \mathop \in J} s_j^{k_j}\) Ring Axioms of $S$
\(\ds \) \(=\) \(\ds \phi \left({f}\right) + \phi \left({g}\right)\) Definition of $\phi$

Therefore $\phi$ preserves addition.


Also:

\(\ds \phi \left({f g}\right)\) \(=\) \(\ds \phi \left({\sum_{k \mathop \in Z} \left({\sum_{p \mathop + q \mathop = k} a_p b_q}\right) \prod_{j \mathop \in J} X_j^{k_j} }\right)\) Definition of Multiplication of Polynomial Forms
\(\ds \) \(=\) \(\ds \sum_{k \mathop \in Z} \psi \left({\sum_{p \mathop + q \mathop = k} a_p b_q}\right) \prod_{j \mathop \in J} s_j^{k_j}\) Definition of $\phi$
\(\ds \) \(=\) \(\ds \sum_{k \mathop \in Z} \left({\sum_{p \mathop + q \mathop = k} \psi \left({a_p}\right) \psi \left({b_q}\right)}\right) \prod_{j \mathop \in J} s_j^{k_j}\) Definition of Ring Homomorphism
\(\ds \) \(=\) \(\ds \left({\sum_{k \mathop \in Z} \left({\psi \left({a_k}\right)}\right) \prod_{j \mathop \in J} s_j^{k_j} }\right) \left({\sum_{k \mathop \in Z} \left({\psi \left({b_k}\right)}\right) \prod_{j \mathop \in J} s_j^{k_j} }\right)\) Ring Axioms of $S$
\(\ds \) \(=\) \(\ds \phi \left({f}\right) \phi \left({g}\right)\) Definition of $\phi$

This shows that $\phi$ is a homomorphism.


Now suppose that $\phi'$ is another such homomorphism.

For each $j \in J$, $\phi'$ must satisfy $\phi'(X_j)=s_j$ and $\phi' \left({r}\right) = \psi \left({r}\right)$ for all $r \in R$.

In addition $\phi'$ must be a homomorphism, so we compute:

\(\ds \phi' \left({\sum_{k \mathop \in Z} a_k \prod_{j \mathop \in J} X_j^{k_j} }\right)\) \(=\) \(\ds \sum_{k \mathop \in Z} \phi' \left({a_k \prod_{j \mathop \in J} X_j^{k_j} }\right)\) $\phi'$ preserves Ring Addition
\(\ds \) \(=\) \(\ds \sum_{k \mathop \in Z} \phi' \left({a_k}\right) \prod_{j \mathop \in J} \phi' \left({X_j}\right)^{k_j}\) $\phi'$ preserves Ring Product
\(\ds \) \(=\) \(\ds \sum_{k \mathop \in Z} \psi \left({a_k}\right) \prod_{j \mathop \in J} s_j^{k_j}\) as $\phi' \left({X_j}\right) = s_j$ and $\phi' \left({r}\right) = \psi \left({r}\right)$ for all $r \in R$

and therefore $\phi' = \phi$.

This concludes the proof.

$\blacksquare$


Remarks

  • The requirement that the rings be commutative is vital. A fundamental difference for polynomials over non-commutative rings is additional difficulty identifying polynomial forms and functions using this method.


Also see