Integral domain

In abstract algebra, an integral domain is a commutative ring with 0 ≠ 1 in which the product of any two non-zero elements is always non-zero; that is, there are no zero divisors. Integral domains are generalizations of the integers and provide a natural setting for studying divisibility.

Related Topics:
Abstract algebra - Commutative ring - Zero divisor - Integer

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Alternatively and equivalently, integral domains may be defined as commutative rings in which the zero ideal {0} is prime, or as the subrings of fields. Viewing the underlying commutative ring as a categorical construction, the previous criterion on zero divisors is equivalent to the condition that every nonzero morphism is a monomorphism (hence also an epimorphism).

Related Topics:
Prime - Subring - Fields - Categorical - Morphism - Monomorphism - Epimorphism

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The condition 0 ≠ 1 only serves to exclude the trivial ring {0} with a single element.

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