Question

Let N1(R) be the sum of all nilpotent ideals in a ring R.
Show that the hypothesis and conclusion in (2) both apply if the ideals in R satisfy DCC.

Answer 1

Assume now ideals in $R$ satisfy $DCC$. Then $N^n = N^{n+1}$ for some $n$. We finish by showing that $M := N^n$ is zero. To see this, assume instead $M <> 0$. Then there exist ideals $A \subseteq M$ with $MAM <> 0$ (for instance $A = M$). Among such ideals $A$, choose a $B$ that is minimal. Then $MBM <> 0$, so $MbM <> 0$ for some $b \in B$. Since $MbM \subseteq B$ and $M(MbM)M = MbM <> 0$, we must have $MbM = B$. In particular, there exists an equation

$$b = \sum_{i=1}^{r} m_i b m_i',$$

where $mi, m_i \in M$.

Now consider the ideal $J \subseteq N$ generated by $\{m_i, m_i'\} : 1 \leq i \leq r$. Since $N$ is a sum of nilpotent ideals, $J$ lies in a finite sum of nilpotent ideals, so $J^k = 0$ for some $k$. Since $b \in JbJ$, we now have by repeated substitution $b \in J^k bJ^k = 0$, a contradiction.