nLab
integral domain

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Definition

A unital ring RR is an integral domain (or simply 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). 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 RR (which includes the empty product 11) 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 RR. 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 RR 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 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.

In constructive mathematics

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

Definition

If we replace “abab is nonzero iff elements aa and bb are nonzero” in the above definition by “abab is nonzero xor either aa or bb 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 010\neq 1.

Such an integral domain DD is ‘discrete’ in that it decomposes as a coproduct D={0}D ×D = \{0\} \sqcup D^\times (where D ×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.

Definition

If we interpret ‘nonzero’ as a reference to a tight apartness relation, thus defining the apartness relation #\# by x#yx # y iff xyx - 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#10\# 1, the condition then means that every element apart from 00 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 Z\mathbf{Z}, Q\mathbf{Q}, Z/n\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.