nLab domain (ring theory)


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



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 RR is a domain if it is nontrivial and the multiplicative submonoid R\{0}R \backslash \{0\} is a cancellative monoid (i.e., 101 \neq 0 and left and right multiplication by cc is injective if c0c \neq 0, which may be combined as left and right multiplication by cc is injective if and only if c0c \neq 0)

By zero divisors

A unital ring RR is an domain if it is nontrivial and has no non-zero zero divisors (i.e., 101 \ne 0 and ab=0a b = 0 implies a=0a = 0 or b=0b = 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 RR (which includes the empty product 11) must be nonzero.


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


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.


In principle, one could just as easily consider a rig or semiring RR. 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 RR 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 RR-algebras are usually not assumed to be associative unital algebras.

See also

Created on December 9, 2022 at 02:32:56. See the history of this page for a list of all contributions to it.