division ring

In particular, the center of a division ring is a field.

Division rings used to be called "fields" in an older usage.

Domains which are not division rings have no minimal right ideals.

More generally, there are no zero divisors in division rings.

Every division ring is therefore a division algebra over its center.

A division ring is a generalization of field, which are not assumed commutative.

Unlike quaternions they do not form a division ring.

This shows that integral domains and division rings don't have such idempotents.

As noted above, all fields are division rings.

P is isomorphic to the projective plane over an alternative division ring.

Restricting the binary operations + and to K, one can shown that is a division ring.

The semigroup of all matrices over a division ring is an epigroup.

Some ternary ring of the plane is an alternative division ring.

The sub division ring and the corresponding subalgebra are each others commutants.

The real and rational quaternions are strictly noncommutative division rings.

A matrix ring over a division ring is semisimple (actually simple).

Many projective geometries can be defined in terms of a coordinate system over a division ring, but others can't.

Division rings differ from fields only in that their multiplication is not required to be commutative.

Vector space: a module where the ring R is a division ring or field.

Neither commutativity nor the division ring assumption is required on the scalars in this case.

Every strongly von Neumann regular ring is a subdirect product of division rings.

Any division ring (including any field) is a near-field.

Suppose that L is a division ring.

One of the best known noncommutative rings is the division ring of quaternions.

Consequently the endomorphism ring of any simple module is a division ring.