Integral domain

An integral domain is a nonzero commutative ring with no nonzero zero-divisors, #m/def/ring i.e. $𝑎 𝑏 = 0$ iff $𝑎 = 0$ or $𝑏 = 0$ . This gives rise the the cancellation property, since all nonzero elements are epic and monic: $𝑎 𝑏 = 𝑎 𝑐$ and $𝑎 \neq 0$ implies $𝑏 = 𝑐$ .

Proof

Since $0 = 𝑎 𝑏 - 𝑎 𝑐 = 𝑎 (𝑏 - 𝑐)$ and $𝑎 \neq 0$ , 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

A finite integral domain is a field
The characteristic of an integral domain is 0 or prime
$𝑅 / 𝐼$ for commutative $𝑅$ is an integral domain iff $𝐼$ is prime
$𝐷 [𝑥]$ is an integral domain iff $𝐷$ is an integral domain
All primes are irreducible in an integral domain

Other results

Krull dimension of an integral domain

Integral domain

Properties

Other results

See also