# nLab domain (ring theory)

Contents

This entry is about domains in ring theory. For other uses, see at domain.

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Definition

Some authors call a domain an integral domain. However, we maintain a distinction between domains and integral domains, and reserve “integral domain” for the commutative domains.

### By the cancellative property

A unital ring $R$ is a domain if it is nontrivial and the multiplicative submonoid $R \backslash \{0\}$ is a cancellative monoid (i.e., $1 \neq 0$ and left and right multiplication by $c$ is injective if $c \neq 0$, which may be combined as left and right multiplication by $c$ is injective if and only if $c \neq 0$)

### By zero divisors

A unital ring $R$ is an domain if it is nontrivial and has no non-zero zero divisors (i.e., $1 \ne 0$ and $a b = 0$ implies $a = 0$ or $b = 0$).

In this definition, the trivial ring is too simple to be an integral domain. You can see this by phrasing this definition without bias as: any product of (finitely many) nonzero elements of $R$ (which includes the empty product $1$) must be nonzero.

## Properties

A domain $R$ is an Ore domain if the set of all nonzero elements is an Ore set in $R$. In that case the Ore localized ring is called the Ore quotient ring? of $R$.

## Examples

For example, the ring of integers, any skewfield, the ring of global sections of the structure sheaf of any integral scheme, an Ore extension of any other domain.

## Generalizations

In principle, one could just as easily consider a rig or semiring $R$. In that case, however, only the definition involving the cancellative property extends to rigs and semirings. Furthermore, we should add the additional requirement that addition in $R$ is cancellable (that is, addition by any element is injective), to make the analogue of the previous paragraph correct. These could be called a domain rig or domain semiring. One could also relax the requirement that the domain be associative or unital, this could be called a domain $\mathbb{Z}$-algebra, in the context where $R$-algebras are usually not assumed to be associative unital algebras.