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Integral domain

An integral domain (or an entire ring) is a commutative ring with a multiplicative identity, say , such that , in which the product of any two non-zero elements is always non-zero.

If and are elements of the integral domain R, we say that divides   or that is a divisor of   or that is a multiple of if there exists an element such that . Symbolically we write , or simply .

The divisibility relation is transitive: if divides and divides , then divides .

An integral domain can be thus defined as a commutative ring with identity and with  no non-zero divisors.

The elements which divide the identity are called the units of . These are precisely the invertible elements in and form a subgroup of the multiplicative semigroup of non-zero elements of . Units divide all other elements.

Moreover, if divides , then divides every multiple of . If divides two elements of , then also divides their sum and difference.

If divides and divides , then we say and are associated elements. Elements and are associated if and only if there exists a unit such that .

If is a non-unit of , we say that is an irreducible element if cannot be written as a product of two non-units of .

If is a non-zero non-unit, we say that is a prime element in if, whenever divides a product , , then divides or divides .

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