Definitions
There are a number of equivalent definitions of integral domain:
- An integral domain is a commutative ring with identity in which the product of any two nonzero elements is not equal to zero.
- An integral domain is a commutative ring with identity in which the zero ideal {0} is a prime ideal.
- An integral domain is a ring with identity that is a subring of a field. This means it is also a commutative ring with identity.
- An integral domain is a commutative ring with identity in which for every non-zero element r, the function that maps every element x of the ring to the product xr is injective. Elements that have this property are called regular, so it is equivalent to require that every non-zero element of the ring be regular.
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