# Nil ideal

A subset $A$ of a ring $R$ is called nil if each element of it is nilpotent (cf. Nilpotent element). An ideal of $R$ is a nil ideal if it is a nil subset. There is a largest nil ideal, which is called the nil radical. One has that
$$\mathop{\rm Jac} ( R) \supset \textrm{ Nil Rad } ( R) \supset \textrm{ Prime Rad } ( R),$$
where $\mathop{\rm Jac} ( R)$ denotes the Jacobson radical of $R$ and $\textrm{ Prime Rad } ( R)$ is the prime radical of $R$, i.e. the intersection of all prime ideals of $R$. Each of the inclusions can be proper. If $R$ is commutative, $\textrm{ Nil Rad } ( R) = \textrm{ Prime Rad } ( R)$. The prime radical is also called the lower nil radical, and the nil radical the upper nil radical.