# Definition:Subdomain

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## 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

## Sources

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