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

This is shown by Polynomials Addition is Associative.

A3

This is shown by Null Polynomial is Additive Identity.

A4

This is shown by Polynomials Have Additive Inverse.

A5

This is shown by Polynomials Addition is Commutative.

Proof of the multiplicative axioms
M1

This is shown by Polynomials Closed under Ring Product.

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.