Question 14
Let $A$ be a ring and let $a$ be an element of $A$. Prove that
a) If $A$ is commutative and $a$ is nilpotent, then $a + x$ is invertible for any invertible element $x \in A$.
b) If $A$ is finite and $a + x$ is invertible for any invertible element $x \in A$, then $a$ is nilpotent.
(An element $a$ of a ring is called nilpotent if there exists some positive integer $n$ such that $a^n = 0$.)
a) If $A$ is commutative and $a$ is nilpotent, then $a + x$ is invertible for any invertible element $x \in A$.
b) If $A$ is finite and $a + x$ is invertible for any invertible element $x \in A$, then $a$ is nilpotent.
(An element $a$ of a ring is called nilpotent if there exists some positive integer $n$ such that $a^n = 0$.)