Question 66
Let $A$ be a ring and let $D$ be the set of all non-invertible elements of $A$. Assuming $a^2 = 0$ for all $a$ in $D$, prove that:
a) $axa = 0$ for all $a$ in $D$ and all $x$ in $A$; and
b) If $D$ is finite and $|D| \ge 2$, there exists $a$ in $D \setminus \{0\}$ such that $ab = ba = 0$ for all $b$ in $D$.
a) $axa = 0$ for all $a$ in $D$ and all $x$ in $A$; and
b) If $D$ is finite and $|D| \ge 2$, there exists $a$ in $D \setminus \{0\}$ such that $ab = ba = 0$ for all $b$ in $D$.