integral domain



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.

In constructive mathematics, one wants to phrase the condition as ab0a b \neq 0 whenever a0a \neq 0 and b0b \neq 0, where \neq is a tight apartness relation relative to which the ring operations are strongly extensional. (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 this is trivial.)

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.

Another equivalent definition is: an integral domain is any subring? of a skewfield. Specifically, any integral domain RR is a subring of its field of fractions.

Some authors require an integral domain to be commutative, even when they do not require this of rings in general. Then they are 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.

Revised on September 15, 2016 11:38:36 by Toby Bartels (