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Difference between revisions of "Integral domain"

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''integral ring''
 
''integral ring''
  
 
A [[Commutative ring|commutative ring]] with identity and without divisors of zero (cf. [[Divisor|Divisor]]). Any field, and also any ring with identity contained in a field, is an integral domain. Conversely, an integral domain can be imbedded in a field. Such an imbedding is given by the construction of the field of fractions (see [[Fractions, ring of|Fractions, ring of]]).
 
A [[Commutative ring|commutative ring]] with identity and without divisors of zero (cf. [[Divisor|Divisor]]). Any field, and also any ring with identity contained in a field, is an integral domain. Conversely, an integral domain can be imbedded in a field. Such an imbedding is given by the construction of the field of fractions (see [[Fractions, ring of|Fractions, ring of]]).
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051390/i0513901.png" /> is an integral domain, then the ring of polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051390/i0513902.png" /> and the ring of formal power series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051390/i0513903.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051390/i0513904.png" /> are also integral domains. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051390/i0513905.png" /> is a commutative ring with identity and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051390/i0513906.png" /> is any ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051390/i0513907.png" />, then the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051390/i0513908.png" /> is an integral domain if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051390/i0513909.png" /> is a [[Prime ideal|prime ideal]]. A ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051390/i05139010.png" /> without nilpotents is an integral domain if and only if the spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051390/i05139011.png" /> is an irreducible topological space (cf. [[Spectrum of a ring|Spectrum of a ring]]).
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If $A$ is an integral domain, then the ring of polynomials $A[X]$ and the ring of formal power series $A[[X]]$ over $A$ are also integral domains. If $A$ is a commutative ring with identity and $I$ is any ideal in $A$, then the ring $A/I$ is an integral domain if and only if $I$ is a [[Prime ideal|prime ideal]]. A ring $A$ without nilpotents is an integral domain if and only if the spectrum of $A$ is an irreducible topological space (cf. [[Spectrum of a ring|Spectrum of a ring]]).
  
Sometimes commutativity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051390/i05139012.png" /> is not required in the definition of an integral domain. Skew-fields and subrings of a skew-field containing the identity are examples of non-commutative integral domains. However, it is not true, in general, that an arbitrary non-commutative integral domain can be imbedded in a skew-field (see [[#References|[2]]], and [[Imbedding of rings|Imbedding of rings]]).
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Sometimes commutativity of $A$ is not required in the definition of an integral domain. Skew-fields and subrings of a skew-field containing the identity are examples of non-commutative integral domains. However, it is not true, in general, that an arbitrary non-commutative integral domain can be imbedded in a skew-field (see [[#References|[2]]], and [[Imbedding of rings|Imbedding of rings]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Lang,  "Algebra" , Addison-Wesley  (1965)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.M. Cohn,  "Free rings and their relations" , Acad. Press  (1985)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C. Faith,  "Algebra: rings, modules, and categories" , '''1''' , Springer  (1973)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Lang,  "Algebra" , Addison-Wesley  (1965)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.M. Cohn,  "Free rings and their relations" , Acad. Press  (1985)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C. Faith,  "Algebra: rings, modules, and categories" , '''1''' , Springer  (1973)</TD></TR></table>

Revision as of 14:53, 15 April 2014

integral ring

A commutative ring with identity and without divisors of zero (cf. Divisor). Any field, and also any ring with identity contained in a field, is an integral domain. Conversely, an integral domain can be imbedded in a field. Such an imbedding is given by the construction of the field of fractions (see Fractions, ring of).

If $A$ is an integral domain, then the ring of polynomials $A[X]$ and the ring of formal power series $A[[X]]$ over $A$ are also integral domains. If $A$ is a commutative ring with identity and $I$ is any ideal in $A$, then the ring $A/I$ is an integral domain if and only if $I$ is a prime ideal. A ring $A$ without nilpotents is an integral domain if and only if the spectrum of $A$ is an irreducible topological space (cf. Spectrum of a ring).

Sometimes commutativity of $A$ is not required in the definition of an integral domain. Skew-fields and subrings of a skew-field containing the identity are examples of non-commutative integral domains. However, it is not true, in general, that an arbitrary non-commutative integral domain can be imbedded in a skew-field (see [2], and Imbedding of rings).

References

[1] S. Lang, "Algebra" , Addison-Wesley (1965)
[2] P.M. Cohn, "Free rings and their relations" , Acad. Press (1985)
[3] C. Faith, "Algebra: rings, modules, and categories" , 1 , Springer (1973)
How to Cite This Entry:
Integral domain. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integral_domain&oldid=15894
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article