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 $ba = 0$.

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