Polynomials Closed under Addition

Theorem

Polynomials over Ring

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

Let $\struct {S, +, \circ}$ be a subring of $R$.

Then $\forall x \in R$, the set $S \sqbrk x$ of polynomials in $x$ over $S$ is closed under the operation $+$.

Polynomial Forms

Let:

$\ds f = \sum_{k \mathop \in Z} a_k \mathbf X^k$

$\ds g = \sum_{k \mathop \in Z} b_k \mathbf X^k$

be polynomials in the indeterminates $\set {X_j: j \in J}$ over the ring $R$.

Then the operation of polynomial addition on $f$ and $g$:

Define the sum:

$\ds f \oplus g = \sum_{k \mathop \in Z} \paren {a_k + b_k} \mathbf X^k$

Then $f \oplus g$ is a polynomial.

That is, the operation of polynomial addition is closed on the set of all polynomials on a given set of indeterminates $\set {X_j: j \in J}$.