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opposite ring (Definition)

If $ R$ 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 $ r_1$ and $ r_2$ in $ R^{op}$ is $ r_2 r_1$.

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

If $ R$ 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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See Also: dual category, non-commutative rings of order four

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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 MSC16B99 (Associative rings and algebras :: General and miscellaneous :: Miscellaneous)
 17A01 (Nonassociative rings and algebras :: General nonassociative rings :: General theory)

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