division ring

Noun

Meaning

division ring (plural division rings)

(algebra) A ring with 0 ≠ 1, such that every non-zero element a has a multiplicative inverse, meaning an element x with ax = xa = 1.

Synonyms

Examples

In fact the converse is also true and this gives a characterization of division rings via their module category: A unital ring R is a division ring if and only if every R- module is free Grillet, Pierre Antoine. Source: Internet

More precisely, a member of the Jacobson radical must project under the canonical homomorphism to the zero of every "right division ring" (each non-zero element of which has a right inverse ) internal to the ring in question. Source: Internet

The transpose of a matrix must be viewed as a matrix over the opposite division ring D op in order for the rule to remain valid. Source: Internet

Any centralizer in a division ring is also a division ring. Source: Internet

It turned out that every finite domain (in particular finite division ring) is a field; in particular commutative (the Wedderburn's little theorem ). Source: Internet

Let K be any division ring (skewfield). Source: Internet

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