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A finite integral domain has multiplicative inverses for all nonzero elements.
Finite integral domains are fields: If is a finite integral domain, then is a field.
Using cancellation in an integral domain, the map is injective for ; finiteness forces it to be surjective, so is a unit. Hence every nonzero element is invertible and the ring is a field.
An integral domain is a commutative ring with such that for all , implies or (equivalently, has no zero divisors).
Cancellation in integral domains: Let be an integral domain and let with . If , then . Likewise, if , then .
In an integral domain, every nonzero element is a regular element (equivalently, there are no nonzero zero divisors), and this is exactly what makes cancellation possible. The same injectivity argument is the starting point for constructing the field of fractions.
Let be a unital ring. An element is a unit if there exists such that .
A field is a commutative ring that is also unital, with , such that every nonzero element is a unit (equivalently, every has a multiplicative inverse).