I am studying about the nilradical $N(R)$ of a unital ring $R$. In my notes, the nilradical of a $R$ is defined as the sum of all nilpotent ideals of $R$.
It says, that $N(R)$ is always a nil ideal, but not a nilpotent.
But there is a lemma that it says that the sum of nilpotent ideals is also a nilpotent ideal. Thus, according to this, $N(R)$ should always be nilpotent ideal.
What do I miss?
The sum of two (or finitely many) nilpotent ideals is nilpotent. However, infinite sums of nilpotent ideals are not necessarily nilpotent.
Consider, however, the ring $$ \Bbb R[x_1,x_2,x_3,\ldots]/(x_1,x_2^2,x_3^3,\ldots) $$ Here the nilradical $(x_1,x_2,x_3,\ldots)$ is indeed nil but not nilpotent.