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 all hold:


 * 1) $$\left({S, + \restriction_S, \circ \restriction_S}\right)$$ is a subring of $$\left({R, +, \circ}\right)$$;
 * 2) 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.