Brown–McCoy radical

Remarks

Equivalently, it is the intersection of the ideals $M$ such that $R/M$ is a simple ring. Example: for $R = \Z$ the maximal ideals are $p\Z$ for primes $p$, and $\bigcap_p p\Z = 0$, so the Brown–McCoy radical of $\Z$ is $0$. Non-example of a maximal ideal: in $\Z$ the ideal $4\Z$ is not maximal, since $4\Z \subsetneq 2\Z \subsetneq \Z$. Every maximal ideal is a primitive ideal, because a simple ring with identity has a faithful simple module; hence the Jacobson radical is contained in the Brown–McCoy radical. The inclusion can be strict. For the ring $R$ of linear maps of a vector space of countably infinite dimension, $J(R) = 0$, while the only maximal ideal is the ideal of maps of finite rank.