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 Z/(2) * Z/(3)? or the ring z/(6)? I would really appreciate the definitions and one or two examples.
What is the order of an element in a ring?
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abstract-algebra
ring-theory
field-theory
finite-fields
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0If the order is small, you can calculate powers until you get the identity. In general you will need to use facts about the ring and the particular element. If you want a full accounting of every possible feature of every possible element of every possible ring, that's not going to happen - you'd need to instead give examples of what kind of rings *you* tend to encounter (e.g. finite fields, matrix algebras, integers mod $n$, direct products thereof, etc.) – 2017-02-01
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0So, until I get the multiplicative identity i.e in Z6 that would be 1 etc.? – 2017-02-01
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0Yes. ${}{}{}{}$ – 2017-02-01
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0ok, what if there's no such m? then order is 0? – 2017-02-01
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0If there's no such $m$, then the order is undefined. Some elements of $\Bbb Z/6$ have this property. – 2017-02-01
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0well, so z/(8) should have an element of order 8. Are you sure I need to take "powers" of the element and mod by 8? it seems that I need to take addition. – 2017-02-01
1 Answers
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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 $a^m=1$.