# A Nil Ideal that is not Nilpotent

## A Nil Ideal that is not Nilpotent

An ideal $I \leq R$ is said to be nil is each of elements is nilpotent. $I$ is said to be nilpotent if there exists some $k$ such that $I^k =0.$ That is, every product of $k$ elements of $I$ is zero.

Consider the ring $R = \mathbb{C}[x_1,x_2,x_3,...]/(x_1,x^2_2,x^3_3,...)$. Then, the ideal $I = (x_1,x_2,x_3,...)$ is nil in $R$ because each of its generators is nilpotent.

Moreover, $I$ is not nilpotent. To see this, assume by contradiction that it is. Then, there exists some $k > 0$ such that $I^k=0.$ This implies that for all $i,$ $x_i^k=0$ in $R$. However, $x_{k+1}^k \neq 0$ in $R$. This is the desired contradiction.