# Definition:Subdomain

## Definition

Let $\struct {R, +, \circ}$ be an algebraic structure with two operations.

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

## Also defined as

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

## 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.