nLab domain (ring theory)

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

---

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.

See also

• integral domain

 External links

• Wikipedia, Domain (ring theory)