Polynomials Closed under Addition/Polynomials over Integral Domain/Proof 2
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Theorem
Let $\struct {R, +, \circ}$ be a commutative ring with unity.
Let $\struct {D, +, \circ}$ be an integral subdomain of $R$.
Then $\forall x \in R$, the set $D \sqbrk x$ of polynomials in $x$ over $D$ is closed under the operation $+$.
Proof
A commutative ring with unity is a ring.
An integral subdomain of a commutative ring with unity $R$ is a subring of $R$.
The result then follows as a special case of Polynomials Closed under Addition: Polynomials over Ring
$\blacksquare$