Ring of Polynomial Forms is Commutative Ring with Unity

Theorem
Let $\left({R, +, \circ}\right)$ be a commutative ring with unity.

Let $A = R \left[{\left\{ {X_j: j \in J}\right\} }\right]$ be the set of all polynomial forms over $R$ in the indeterminates $\left\{{X_j: j \in J}\right\}$.

Then $\left({A, +, \circ}\right)$ is a commutative ring with unity.

Proof
We must show that the commutative and unitary ring axioms are satisfied:

Proof of the additive axioms
A1:

This is shown by Polynomials Closed under Addition.

A2-A5:

According to the formal definition, a polynomial is a map from the free commutative monoid to $R$.

Now observe that addition of polynomial forms is induced by addition in $R$.

Therefore:


 * A2 is shown by Structure Induced by Associative Operation is Associative


 * A3 is shown by Induced Structure Identity


 * A4 is shown by Induced Structure Inverse


 * A5 is shown by Structure Induced by Commutative Operation is Commutative

Proof of the multiplicative axioms
M1:

This is shown by Polynomials Closed under Ring Product.

Multiplication of polynomial forms is not induced by multiplication in $R$, so we must show the multiplicative axioms by hand.

M2:

This is shown by Multiplication of Polynomials is Associative.

M3:

This is shown by Polynomials Contain Multiplicative Identity.

M4:

This is shown by Multiplication of Polynomials is Commutative.

D:

This is shown by Multiplication of Polynomials Distributes over Addition.

Therefore, all of the axioms of a commutative ring with unity are satisfied.