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   polynomials over $R$ in  the indeterminates  $\{{X_j:j\in J}\}$.

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

For  two polynomials $\displaystyle f = \sum_{k\in Z} a_k \mathbf  X^k,\ g =  \sum_{k\in Z} b_k \mathbf X^k$ in $A$, define the sum:


 * $\displaystyle f + g = \sum_{k\in Z} \left({a_k + b_k}\right)\mathbf X^k$

and the product


 * $\displaystyle f \circ g = \sum_{k\in Z} c_i \mathbf X^k$

where $\displaystyle c_k = \sum_{j+l = k} f \left({\mathbf  X^j}\right) \circ g \left({\mathbf X^l}\right)$.

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

Proof
We must show that the following axioms are satisfied:

Multiplicative axioms
Let


 * $\displaystyle f = \sum_{k\in Z} a_k \mathbf X^k$


 * $g = \sum_{k\in Z} b_k \mathbf X^k$


 * $h = \sum_{k\in Z} c_k \mathbf X^k$ be arbitrary elements of $A$.

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.