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A [[Ring|ring]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074540/p0745401.png" /> in which the product of two two-sided ideals (cf. [[Ideal|Ideal]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074540/p0745402.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074540/p0745403.png" /> is equal to the zero ideal if and only if either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074540/p0745404.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074540/p0745405.png" /> is the zero ideal. In other words, the ideals of a prime ring form a [[Semi-group|semi-group]] without zero divisors under multiplication. A ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074540/p0745406.png" /> is a prime ring if and only if the right (left) [[Annihilator|annihilator]] of any non-zero right (correspondingly, left) ideal of it is equal to , and also if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074540/p0745407.png" /> for any non-zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074540/p0745408.png" />. The centre of a prime ring is an [[Integral domain|integral domain]]. Any [[Primitive ring|primitive ring]] is prime. If a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074540/p0745409.png" /> does not contain non-zero nil ideals, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074540/p07454010.png" /> is the subdirect sum of prime rings. The class of prime rings plays an important part in the theory of radicals of rings (cf. [[Radical of rings and algebras|Radical of rings and algebras]]) [[#References|[1]]].
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A [[Ring|ring]] $R$ in which the product of two two-sided [[ideal]]s $P$ and $Q$ is equal to the zero ideal if and only if either $P$ or $Q$ is the zero ideal. In other words, the ideals of a prime ring form a [[semi-group]] without zero divisors under multiplication. A ring $R$ is a prime ring if and only if the right (left) [[annihilator]] of any non-zero right (correspondingly, left) ideal is equal to $0$, and also if and only if $aRb\ne0$ for any non-zero $a,b\in R$. The [[Centre of a ring|centre]] of a prime ring is an [[integral domain]]. Any [[primitive ring]] is prime. If a ring $R$ does not contain non-zero nil ideals, then $R$ is the subdirect sum of prime rings. The class of prime rings plays an important part in the theory of radicals of rings (cf. [[Radical of rings and algebras]]) [[#References|[1]]].
  
There is the following generalization of the concept of a prime ring. A ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074540/p07454011.png" /> is said to be semi-prime if it does not have non-zero nilpotent ideals.
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There is the following generalization of the concept of a prime ring. A ring $R$ is said to be '''semi-prime''' if it does not have non-zero nilpotent ideals.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.A. Andrunakievich,  Yu.M. Ryabukhin,  "Radicals of algebras and lattice theory" , Moscow  (1979)  (In Russian)</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">  I.N. Herstein,  "Noncommutative rings" , Math. Assoc. Amer.  (1968)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  V.A. Andrunakievich,  Yu.M. Ryabukhin,  "Radicals of algebras and lattice theory" , Moscow  (1979)  (In Russian)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  N. Jacobson,  "Structure of rings" , Amer. Math. Soc.  (1956)</TD></TR>
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<TR><TD valign="top">[3]</TD> <TD valign="top">  I.N. Herstein,  "Noncommutative rings" , Math. Assoc. Amer.  (1968)</TD></TR></table>

Latest revision as of 18:30, 25 September 2016

A ring $R$ in which the product of two two-sided ideals $P$ and $Q$ is equal to the zero ideal if and only if either $P$ or $Q$ is the zero ideal. In other words, the ideals of a prime ring form a semi-group without zero divisors under multiplication. A ring $R$ is a prime ring if and only if the right (left) annihilator of any non-zero right (correspondingly, left) ideal is equal to $0$, and also if and only if $aRb\ne0$ for any non-zero $a,b\in R$. The centre of a prime ring is an integral domain. Any primitive ring is prime. If a ring $R$ does not contain non-zero nil ideals, then $R$ is the subdirect sum of prime rings. The class of prime rings plays an important part in the theory of radicals of rings (cf. Radical of rings and algebras) [1].

There is the following generalization of the concept of a prime ring. A ring $R$ is said to be semi-prime if it does not have non-zero nilpotent ideals.

References

[1] V.A. Andrunakievich, Yu.M. Ryabukhin, "Radicals of algebras and lattice theory" , Moscow (1979) (In Russian)
[2] N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956)
[3] I.N. Herstein, "Noncommutative rings" , Math. Assoc. Amer. (1968)
How to Cite This Entry:
Prime ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Prime_ring&oldid=16729
This article was adapted from an original article by K.A. Zhevlakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article