# Subdomain Test

## Theorem

Let $S$ be a subset of an integral domain $\struct {R, +, \circ}$.

Then $\struct {S, +\restriction_S, \circ \restriction_S}$ is a subdomain of $\struct {R, +, \circ}$ if and only if these conditions hold:

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

## Proof

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

$\blacksquare$