Hw12 Problem 13
Return to Homework 12, Glossary, Theorems
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)\begin{equation} ba = (ba) ^ {n} \end{equation}
(2)
\begin{align} = b(\underbrace{abab...abab}_{n-1\ terms})a. \end{align}
Then, since $ab = 0$,
(3)\begin{equation} ba = b(0)a \end{equation}
(4)
\begin{equation} = 0 \end{equation}
Therefore in a ring in which $x ^ {n} = x$ for all $x$, $ab = 0$ implies $ba = 0$.