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A two-sided [[Ideal|ideal]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074520/p0745201.png" /> of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074520/p0745202.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074520/p0745203.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074520/p0745204.png" /> are ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074520/p0745205.png" />, implies either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074520/p0745206.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074520/p0745207.png" />. For an associative ring the following is an equivalent definition in terms of elements:
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074520/p0745208.png" /></td> </tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074520/p0745209.png" /> are elements in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074520/p07452010.png" />. Every [[Primitive ideal|primitive ideal]] is a prime ideal.
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A two-sided [[Ideal|ideal]] $  P $
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of a ring  $  R $
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such that  $  AB \subseteq P $,
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where  $  A , B $
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are ideals of  $  R $,
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implies either  $  A \subseteq P $
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or  $  B \subseteq P $.  
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For an associative ring the following is an equivalent definition in terms of elements:
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074520/p07452011.png" /> be an associative-commutative ring with an identity. Then an ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074520/p07452012.png" /> is prime if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074520/p07452013.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074520/p07452014.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074520/p07452015.png" />, i.e. if and only if the quotient ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074520/p07452016.png" /> is an [[Integral domain|integral domain]]. In this case every maximal ideal is prime and the intersection of all prime ideals is the radical of the null ideal (i.e. is the set of nilpotent elements).
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$$
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a R b \subseteq P  \Rightarrow  a \in P  \textrm{ or }  b \in P ,
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$$
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where  $  a , b $
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are elements in  $  R $.  
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Every [[Primitive ideal|primitive ideal]] is a prime ideal.
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 +
Let  $  R $
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be an associative-commutative ring with an identity. Then an ideal $  P \subset  R $
 +
is prime if and only if $  ab \in P $
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implies $  a \in P $
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or $  b \in P $,  
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i.e. if and only if the quotient ring $  R / P $
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is an [[Integral domain|integral domain]]. In this case every maximal ideal is prime and the intersection of all prime ideals is the radical of the null ideal (i.e. is the set of nilpotent elements).
  
 
A generalization of the concept of a prime ideal is that of a [[Primary ideal|primary ideal]]. In the theory of [[Primary decomposition|primary decomposition]], the prime ideals play the same role as the prime numbers do in the decomposition of integers in powers of prime numbers, while the primary ideals play the role of powers of prime numbers.
 
A generalization of the concept of a prime ideal is that of a [[Primary ideal|primary ideal]]. In the theory of [[Primary decomposition|primary decomposition]], the prime ideals play the same role as the prime numbers do in the decomposition of integers in powers of prime numbers, while the primary ideals play the role of powers of prime numbers.
  
An ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074520/p07452017.png" /> in a [[Lattice|lattice]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074520/p07452018.png" /> is called prime if
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An ideal $  P $
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in a [[Lattice|lattice]] $  L $
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is called prime if
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074520/p07452019.png" /></td> </tr></table>
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$$
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ab \in P  \Rightarrow  a \in P  \textrm{ or }  b \in P .
 +
$$
  
An ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074520/p07452020.png" /> is prime if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074520/p07452021.png" /> is a prime filter, i.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074520/p07452022.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074520/p07452023.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074520/p07452024.png" />.
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An ideal $  P $
 +
is prime if and only if $  F = L \setminus  P $
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is a prime filter, i.e. if $  a+ b \in F $
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implies $  a \in F $
 +
or $  b \in F $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Commutative algebra" , Addison-Wesley  (1972)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Jacobson,  "Structure of rings" , Amer. Math. Soc.  (1956)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  O. Zariski,  P. Samuel,  "Commutative algebra" , '''1''' , Springer  (1975)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L.A. Skornyakov,  "Elements of lattice theory" , A. Hilger  (1977)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Commutative algebra" , Addison-Wesley  (1972)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Jacobson,  "Structure of rings" , Amer. Math. Soc.  (1956)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  O. Zariski,  P. Samuel,  "Commutative algebra" , '''1''' , Springer  (1975)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L.A. Skornyakov,  "Elements of lattice theory" , A. Hilger  (1977)  (Translated from Russian)</TD></TR></table>

Latest revision as of 08:07, 6 June 2020


A two-sided ideal $ P $ of a ring $ R $ such that $ AB \subseteq P $, where $ A , B $ are ideals of $ R $, implies either $ A \subseteq P $ or $ B \subseteq P $. For an associative ring the following is an equivalent definition in terms of elements:

$$ a R b \subseteq P \Rightarrow a \in P \textrm{ or } b \in P , $$

where $ a , b $ are elements in $ R $. Every primitive ideal is a prime ideal.

Let $ R $ be an associative-commutative ring with an identity. Then an ideal $ P \subset R $ is prime if and only if $ ab \in P $ implies $ a \in P $ or $ b \in P $, i.e. if and only if the quotient ring $ R / P $ is an integral domain. In this case every maximal ideal is prime and the intersection of all prime ideals is the radical of the null ideal (i.e. is the set of nilpotent elements).

A generalization of the concept of a prime ideal is that of a primary ideal. In the theory of primary decomposition, the prime ideals play the same role as the prime numbers do in the decomposition of integers in powers of prime numbers, while the primary ideals play the role of powers of prime numbers.

An ideal $ P $ in a lattice $ L $ is called prime if

$$ ab \in P \Rightarrow a \in P \textrm{ or } b \in P . $$

An ideal $ P $ is prime if and only if $ F = L \setminus P $ is a prime filter, i.e. if $ a+ b \in F $ implies $ a \in F $ or $ b \in F $.

References

[1] N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)
[2] N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956)
[3] O. Zariski, P. Samuel, "Commutative algebra" , 1 , Springer (1975)
[4] L.A. Skornyakov, "Elements of lattice theory" , A. Hilger (1977) (Translated from Russian)
How to Cite This Entry:
Prime ideal(2). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Prime_ideal(2)&oldid=11807
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article