Integral domain

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In abstract algebra, an integral domain is a commutative ring with 1 ≠ 0 (i.e. the multiplicative identity is not equal to the additive identity) that has no zero divisors[1][2][3][4][5]. Integral domains are generalizations of the integers and provide a natural setting for studying divisibility. An integral domain is a commutative domain with identity.[6][7][8]

Viewing the underlying commutative ring as a preadditive category, the above criterion on zero divisors is equivalent to the condition that every nonzero morphism is a monomorphism (hence also an epimorphism, by making use of the bilinear structure on the set of morphisms).

The condition 1 ≠ 0 only serves to exclude the trivial ring {0}.

The above is how "integral domain" is almost universally defined, but there is some variation. In particular, noncommutative integral domains are sometimes admitted,[9] and very rarely the condition 1 ≠ 0 is omitted.[10] However, we follow the much more usual convention of reserving the term integral domain for the commutative case and use domain for the noncommutative case. Some sources, notably Lang, use the term entire ring for integral domain.[11]

Some specific kinds of integral domains are given with the following chain of class inclusions:

Contents

Definitions

There are a number of equivalent definitions of integral domain:

  • An integral domain is a commutative ring with identity such that for any two elements a and b of the ring, ab = 0 implies either a = 0 or b = 0.
  • An integral domain is a commutative ring with identity in which the zero ideal {0} is a prime ideal.
  • An integral domain is a commutative ring with identity that is a subring of a field.
  • An integral domain is a commutative ring with identity such that for every non-zero element r of the ring, the function that maps every element x of the ring to the product xr is injective. Elements which have this property are called regular, so it is equivalent to require that every non-zero element of the ring is regular.

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