Subdomain Test

Theorem
Let $S$ be a subset of an integral domain $\left({R, +, \circ}\right)$.

Then $\left({S, +\restriction_S, \circ \restriction_S}\right)$ is a subdomain of $\left({R, +, \circ}\right)$ iff these conditions hold:


 * $(1): \quad$ $\left({S, + \restriction_S, \circ \restriction_S}\right)$ is a subring of $\left({R, +, \circ}\right)$
 * $(2): \quad$ The unity of $R$ is also in $S$, i.e. $1_R = 1_S$.

Proof
By Idempotent Elements of Ring with No Zero Divisors, it follows that the unity of a subdomain is the unity of the integral domain it's a subdomain of.