View source for Prime ring

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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 $R$ is said to be '''semi-prime''' if it does not have non-zero nilpotent ideals.

====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>