# Ring of Polynomial Forms is Commutative Ring with Unity

## Theorem

Let $\struct {R, +, \circ}$ be a commutative ring with unity.

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

Then $\struct {A, +, \circ}$ is a commutative ring with unity.

## Proof

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

A commutative and unitary ring is an algebraic structure $\struct {R, *, \circ}$, on which are defined two binary operations $\circ$ and $*$, which satisfy the following conditions:

\((\text A 0)\) | $:$ | Closure under addition | \(\displaystyle \forall a, b \in R:\) | \(\displaystyle a * b \in R \) | ||||

\((\text A 1)\) | $:$ | Associativity of addition | \(\displaystyle \forall a, b, c \in R:\) | \(\displaystyle \paren {a * b} * c = a * \paren {b * c} \) | ||||

\((\text A 2)\) | $:$ | Commutativity of addition | \(\displaystyle \forall a, b \in R:\) | \(\displaystyle a * b = b * a \) | ||||

\((\text A 3)\) | $:$ | Identity element for addition: the zero | \(\displaystyle \exists 0_R \in R: \forall a \in R:\) | \(\displaystyle a * 0_R = a = 0_R * a \) | ||||

\((\text A 4)\) | $:$ | Inverse elements for addition: negative elements | \(\displaystyle \forall a \in R: \exists a' \in R:\) | \(\displaystyle a * a' = 0_R = a' * a \) | ||||

\((\text M 0)\) | $:$ | Closure under product | \(\displaystyle \forall a, b \in R:\) | \(\displaystyle a \circ b \in R \) | ||||

\((\text M 1)\) | $:$ | Associativity of product | \(\displaystyle \forall a, b, c \in R:\) | \(\displaystyle \paren {a \circ b} \circ c = a \circ \paren {b \circ c} \) | ||||

\((\text M 2)\) | $:$ | Commutativity of product | \(\displaystyle \forall a, b \in R:\) | \(\displaystyle a \circ b = b \circ a \) | ||||

\((\text M 3)\) | $:$ | Identity element for product: the unity | \(\displaystyle \exists 1_R \in R: \forall a \in R:\) | \(\displaystyle a \circ 1_R = a = 1_R \circ a \) | ||||

\((\text D)\) | $:$ | Product is distributive over addition | \(\displaystyle \forall a, b, c \in R:\) | \(\displaystyle a \circ \paren {b * c} = \paren {a \circ b} * \paren {a \circ c} \) | ||||

\(\displaystyle \paren {a * b} \circ c = \paren {a \circ c} * \paren {b \circ c} \) |

These criteria are called the **commutative and unitary ring axioms**.

### 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.

$\blacksquare$