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Prime ring

From Encyclopedia of Mathematics
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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://www.encyclopediaofmath.org/index.php?title=Prime_ring&oldid=39307
This article was adapted from an original article by K.A. Zhevlakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article