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A commutative ring is a ring where the multiplication operation is commutative.
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In a ring with unity
, the distributive laws show that for -
(Fraleigh e18.38) In a commutative ring:
In fact, the converse is also true if
the above holds, then the ring is commutative -
In a commutative ring, the usual binomial expansion for
holds -
(Fraleigh e18.44a) The set of all idempotent elements in a commutative ring is closed under ring multiplication
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(Fraleigh e18.46) If
and are nilpotent elements of a commutative ring, then is also nilpotent In fact if
, such that . Then