Ring of Polynomial Forms

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:

Additive axioms

A1:	$A$ is closed under addition;
A2:	Addition in $A$ is associative;
A3:	$A$ contains an additive identity;
A4:	Each $f \in A$ has an additive inverse;
A5:	Addition in $A$ is commutative;

Multiplicative axioms

M1:	$A$ is closed under multiplication;
M2:	Multiplication in $A$ is associative;
M3:	$A$ contains a multiplicative identity;
M4:	Multiplication in $A$ is commutative;
D:	Multiplication distributes over addition in $A$.

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.

$\blacksquare$