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integral domain

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Definition

By the cancellative property

A unital ring RR is an integral domain (or simply 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)

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.

By zero divisors

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).

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.

In constructive mathematics

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

Discrete integral domains

Definition

If we replace “left and right multiplication by cc is injective iff cc is nonzero” in the above definition by “left and right multiplication by cc is injective xor cc 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 submomoid of DD that is cancellative). 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.

Heyting integral domains

Definition

If we interpret ‘nonzero’ as a reference to a tight apartness relation #\#, thus defining the apartness relation #\# by x#yx#y if left and right multiplication by xyx - y is injective, then we obtain the notion of Heyting integral domain. (As shown here, the ring operations become strongly extensional functions). It follows 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.

Properties

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.

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 integral domain.

Generalizations

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. 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.

Last revised on May 21, 2021 at 14:47:23. See the history of this page for a list of all contributions to it.