Definition:Polynomial Ring/Monoid Ring on Free Monoid on Set

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

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

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

Let $+$ and $\circ$ denote the standard addition and multiplication of polynomial forms.

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

Proof
We must show that the following 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 Induced Structure Associative


 * A3 is shown by Induced Structure Identity


 * A4 is shown by Induced Structure Inverse


 * A5 is shown by Induced Structure 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.