Ring

Integral domain

An integral domain is a nonzero commutative ring with no nonzero zero-divisors, ring i.e. 𝑎𝑏 =0 iff 𝑎 =0 or 𝑏 =0. This gives rise the the cancellation property, since all nonzero elements are epic and monic: 𝑎𝑏 =𝑎𝑐 and 𝑎 ≠0 implies 𝑏 =𝑐.

Proof

Since 0 =𝑎𝑏 −𝑎𝑐 =𝑎(𝑏 −𝑐) and 𝑎 ≠0, it follows 𝑏 −𝑐 =0 and hence 𝑏 =𝑐.

Note that by moving to the Field of fractions we can get cancellation in the normal way.

Properties

1. A finite integral domain is a field
2. The characteristic of an integral domain is 0 or prime
3. Condition for a quotient commutative ring to be an integral domain
4. The polynomial ring over an integral domain is an integral domain
5. All primes are irreducible in an integral domain

Other results

• Krull dimension of an integral domain

See also

• Field of fractions
• Euclidean domain

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