PlanetMath: ideal of elements with finite order

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ideal of elements with finite order

(Theorem)

Theorem. The set of all elements of a ring, which have a finite order in the additive group of the ring, is a (two-sided) ideal of the ring.

Proof. Let be the set of the elements with finite order in the ring . Denote by the order of . Take arbitrary elements $a,\,b$ of the set .

If ${\mathrm{lcm}}(o(a),\,o(b)) = n = ko(a) = lo(b)$ , then

$\displaystyle n(a-b) = na-nb = ko(a)a-lo(b)b = k\cdot0-l\cdot0 = 0-0 = 0.$

Thus $o(a-b) \leqq n < \infty$ and so $a-b \in S$ .

For any element of we have

$\displaystyle o(a)(ra) = \underbrace{ra+ra+\ldots+ra}_{o(a)} = r(\underbrace{a+a+\ldots+a}_{o(a)}) = r(o(a)a) = r\cdot0 = 0.$

Therefore, $o(ra) \leqq o(a) < \infty$ and $ra \in S$ . Similarly, $ar \in S$ .

Since satisfies the conditions for an ideal, the theorem has been proven.

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Cross-references: ideal, additive group, order, finite, ring

This is version 5 of ideal of elements with finite order, born on 2008-03-01, modified 2008-03-05.
Object id is 10355, canonical name is IdealOfElementsWithFiniteOrder.
Accessed 186 times total.

Classification:

AMS MSC:	16D25 (Associative rings and algebras :: Modules, bimodules and ideals :: Ideals)
	20A05 (Group theory and generalizations :: Foundations :: Axiomatics and elementary properties)

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