Nilpotent ideal

Remarks

Every nilpotent ideal is a nil ideal, since $I^n = 0$ gives $a^n = 0$ for all $a \in I$. Example: let $R$ be the ring of upper triangular $2 \times 2$ matrices over a field $K$ and let $I$ be the set of its matrices with zero diagonal. $I$ is a two-sided ideal, and the product of two matrices of the form $\begin{pmatrix} 0 & x \\ 0 & 0 \end{pmatrix}$ is zero, so $I^2 = 0$. Non-example: let $R = \Z[x_1, x_2, \dots] / (x_1, x_2^2, x_3^3, \dots)$ and let $I$ be the ideal generated by the images of all $x_k$. $I$ is nil, since it is generated by nilpotent elements of a commutative ring. It is not nilpotent, because for every $n$ the product $x_{n+1}^{n}$ of $n$ elements of $I$ is nonzero in $R$.