Hw12 Problem 13

Problem: (*) Let $n$ be an integer greater than $1$. In a ring in which $x ^ {n} = x$ for all $x$, show that $ab = 0$ implies $ba = 0$.

Solution: Because $x ^ {n} = x$,

(1)
$$ba = (ba) ^ {n}$$
(2)
\begin{align} = b(\underbrace{abab...abab}_{n-1\ terms})a. \end{align}

Then, since $ab = 0$,

(3)
$$ba = b(0)a$$
(4)
$$= 0$$

Therefore in a ring in which $x ^ {n} = x$ for all $x$, $ab = 0$ implies $ba = 0$.