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.

Also see

• Subdomain Test

Sources

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