PlanetMath: opposite ring

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opposite ring

(Definition)

If is a ring, then we may construct the opposite ring $R^{op}$ which has the same underlying abelian group structure, but with multiplication in the opposite order: the product of and in $R^{op}$ is .

If is a left -module, then it can be made into a right $R^{op}$ -module, where a module element , when multiplied on the right by an element of $R^{op}$ , yields the that we have with our left -module action on . Similarly, right -modules can be made into left $R^{op}$ -modules.

If is a commutative ring, then it is equal to its own opposite ring.

Similar constructions occur in the opposite group and opposite category.

"opposite ring" is owned by antizeus.

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Cross-references: opposite category, opposite group, occur in, similar, commutative ring, action, module, right, product, order, opposite, multiplication, structure, abelian group, ring
There are 3 references to this entry.

This is version 2 of opposite ring, born on 2001-10-20, modified 2004-03-11.
Object id is 416, canonical name is OppositeRing.
Accessed 2297 times total.

Classification:

AMS MSC:	16B99 (Associative rings and algebras :: General and miscellaneous :: Miscellaneous)
	17A01 (Nonassociative rings and algebras :: General nonassociative rings :: General theory)

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