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.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $21$. Rings and Integral Domains: Theorem $21.4$