Definition:Subdomain

Definition

Let $\left({R, +, \circ}\right)$ be an algebraic structure with two operations.

A subdomain of $\left({R, +, \circ}\right)$ is a subset $S$ of $R$ such that $\left({S, +_S, \circ_S}\right)$ is an integral domain.

Also known as

Some sources prefer to call this an integral subdomain for clarity of exposition. $\mathsf{Pr} \infty \mathsf{fWiki}$ adopts this convention as appropriate.

Also see

• Subdomain Test

Notes

Some sources insist that $R$ must be a ring for $S$ to be definable as a subdomain, but this limitation can be too restricting.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): $\S 21$