# Prime ring

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