More explicitly a ring, , is an integral domain if:
- it is commutative,
- (where and are the additive and multiplicative identities, respectively)
- and it contains no zero divisors (i.e. there are no nonzero such that ).
Some common examples of integral domains are:
- The ring of integers.
- Any field.
- The p-adic integers, .
- For any integral domain, , the polynomial ring is also an integral domain.
- Any finite integral domain is a field.
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