Generally speaking, how can I determine the order of an element in a ring? I have background in group theory, and there, the order of an element $e$ is the smallest number $m$ such that $e^m$ is the identity. How can I calculate the order of elements in for example the ring $\mathbb{Z}/(2) \times \mathbb{Z}/(3)$? or the ring $\mathbb{Z}/(6)$? I would really appreciate the definitions and one or two examples.
2026-04-04 06:01:39.1775282499
What is the order of an element in a ring?
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As far as I know, when people say "the order of a ring element" they usually mean additive order, i.e. the smallest positive integer $m$ such that $m \cdot a =0$. However, sometimes (as it is in the case of $\mathbb{Z}_n$) by the order of a ring element one can mean the "multiplicative order," i.e. the smallest positive integer $m$ such that $a^m=1$.