Question 12
Let $m$ be a positive integer and $p$ a prime. Suppose $A$ is a unitary ring having exactly $m$ invertible elements and such that $\underbrace{1+1+\cdots+1}_{p \text{ times}} = 0$.
Prove that $A$ has non-zero nilpotent elements if and only if $p$ divides $m$.
Prove that $A$ has non-zero nilpotent elements if and only if $p$ divides $m$.