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integral domain

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Definition

A unital ring is an integral domain (or simply domain) if it has no zero divisors (i.e. ab=0a b = 0 implies either 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 none of this matters.)

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 July 19, 2014 01:38:54 by Urs Schreiber (82.113.106.79)