Ring

Integral domain

An integral domain is a nonzero commutative ring where every element apart from zero is regular. Equivalently it supports the null factor law

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. 𝑅/𝐼 for commutative 𝑅 is an integral domain iff 𝐼 is prime
4. 𝐷[𝑥] is an integral domain iff 𝐷 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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