# 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 $$ord(ab)=n?$$

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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.