integral domain

A unital ring $R$ is an **integral domain** (or simply 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 $b = 0$). 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 integral domain.

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

Some authors require an integral domain to be commutative, even when they do not require this of rings in general. Commutative integral domains are precisely subrings of fields.

In principle, one could just as easily consider a rig or semiring $R$. In that case, however, a zero divisor is not what the name literally implies: the definition is that multiplication by a nonzero element (on either side) is injective. 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. Since ‘integral domain’ is too specific and ‘integral ring’ is not standard (and means something else in the phrase ‘integral ring extension’), it's not clear exactly what these should be called; perhaps *integral cancellable rig/semiring* is sufficiently unambiguous.

An integral 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 $R$.

in constructive mathematics, there are different inequivalent ways to define an integral domain.

If we replace “$ab$ is nonzero iff elements $a$ and $b$ are nonzero” in the above definition by “$ab$ is nonzero xor either $a$ or $b$ is zero” (which is equivalent in classical logic but stronger in constructive logic), then we obtain the notion of **discrete integral domain**. This condition implies that $0\neq 1$.

Such an integral domain $D$ is ‘discrete’ in that it decomposes as a coproduct $D = \{0\} \sqcup D^\times$ (where $D^\times$ is the subset of elements that are not zero-divisors). An advantage is that this is a coherent theory and hence also a geometric theory. A disadvantage is that this axiom is not satisfied (constructively) by the ring of real numbers (however these are defined), although it is satisfied by the ring of integers and the ring of rationals.

If we interpret ‘nonzero’ as a reference to a tight apartness relation, thus defining the apartness relation $\#$ by $x # y$ iff $x - y$ is invertible, then we obtain the notion of **Heyting integral domain**. (As shown here, the ring operations become strongly extensional functions.) In addition to $0\# 1$, the condition then means that every element apart from $0$ is not a zero-divisor.

This is how ‘practising’ constructive analysts of the Bishop school usually define the simple word ‘integral domain’. An advantage is that the (located Dedekind) real numbers form a Heyting integral domain. A disadvantage is that this is not a coherent axiom and so cannot be internalized in as many categories.

Of course, if the underlying set of the ring has decidable equality —as is true of $\mathbf{Z}$, $\mathbf{Q}$, $\mathbf{Z}/n$, finite fields, etc— then a Heyting integral domain is a discrete integral domain.

Last revised on April 3, 2021 at 13:58:20. See the history of this page for a list of all contributions to it.