User:Linus44/Sandbox

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.