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 | \(\ds \forall a, b \in R:\) | \(\ds a * b \in R \) | |||||
\((\text A 1)\) | $:$ | Associativity of addition | \(\ds \forall a, b, c \in R:\) | \(\ds \paren {a * b} * c = a * \paren {b * c} \) | |||||
\((\text A 2)\) | $:$ | Commutativity of addition | \(\ds \forall a, b \in R:\) | \(\ds a * b = b * a \) | |||||
\((\text A 3)\) | $:$ | Identity element for addition: the zero | \(\ds \exists 0_R \in R: \forall a \in R:\) | \(\ds a * 0_R = a = 0_R * a \) | |||||
\((\text A 4)\) | $:$ | Inverse elements for addition: negative elements | \(\ds \forall a \in R: \exists a' \in R:\) | \(\ds a * a' = 0_R = a' * a \) | |||||
\((\text M 0)\) | $:$ | Closure under product | \(\ds \forall a, b \in R:\) | \(\ds a \circ b \in R \) | |||||
\((\text M 1)\) | $:$ | Associativity of product | \(\ds \forall a, b, c \in R:\) | \(\ds \paren {a \circ b} \circ c = a \circ \paren {b \circ c} \) | |||||
\((\text M 2)\) | $:$ | Commutativity of product | \(\ds \forall a, b \in R:\) | \(\ds a \circ b = b \circ a \) | |||||
\((\text M 3)\) | $:$ | Identity element for product: the unity | \(\ds \exists 1_R \in R: \forall a \in R:\) | \(\ds a \circ 1_R = a = 1_R \circ a \) | |||||
\((\text D)\) | $:$ | Product is distributive over addition | \(\ds \forall a, b, c \in R:\) | \(\ds a \circ \paren {b * c} = \paren {a \circ b} * \paren {a \circ c} \) | |||||
\(\ds \paren {a * b} \circ c = \paren {a \circ c} * \paren {b \circ c} \) |
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 Pointwise Inverse in Induced Structure
- 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.
$\blacksquare$