The order of a product of two elements in a ring

by LipCaty   Last Updated July 17, 2019 13:20 PM

Let $(R,+,\cdot)$ be a non-commutative ring with zero $0$ and identity $1$. $a,b\in (R,+)$ are two elements of finite order $m,n$ respectively; in notations, $ord(a)=m,ord(n)=n$. Then what is the order of the product $ab\in (R,+)$ ? In particular, if $m=n$, then is it ture that $ord(ab)=n?$


Answers 1

Basically, it can be anything. Consider the ring of the $2\times2$ real matrices. For instance, if$$a=\begin{bmatrix}2 & 1 \\ -3 & -2\end{bmatrix}\text{ and }b=\begin{bmatrix}-1 & 0 \\ 0 & 1\end{bmatrix},$$then both $a$ and $b$ have order $2$, but$$a.b=\begin{bmatrix}-2&1\\3&-2\end{bmatrix},$$whose order is infinity.

José Carlos Santos
José Carlos Santos
July 17, 2019 13:18 PM

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