In algebra, a division ring, also called a skew field (or, occasionally, a sfield), is a nontrivial ring in which division by nonzero elements is defined...

Saturn's ring was composed of multiple smaller rings with gaps between them; the largest of these gaps was later named the Cassini Division. This division is...

A division ring is a ring such that every non-zero element is a unit. A commutative division ring is a field. A prominent example of a division ring that...

different language, modules; special classes of rings (group rings, division rings, universal enveloping algebras); related structures like rngs; as well...

Equivalently, a noncommutative ring is a ring that is not a commutative ring. Noncommutative algebra is the part of ring theory devoted to study of properties...

normed division algebras are R, C, H, and the (non-associative) algebra O. Pontryagin variant. If D is a connected, locally compact division ring, then...

simple ring is a non-zero ring that has no two-sided ideal besides the zero ideal and itself. In particular, a commutative ring is a simple ring if and...

Shestakov, and Shirshov. The projective plane over any alternative division ring is a Moufang plane. Every composition algebra is an alternative algebra...

leads to the concept of a division ring or skew field; sometimes associativity is weakened as well. Historically, division rings were sometimes referred...

ring need not be a field or division ring, and there are many projective planes that are not constructed from a division ring. They are called non-Desarguesian...

application in projective geometry requires that the scalars come from a division ring (skew field), K, and this means that the "vectors" should be replaced...

nonzero ring R in which every nonzero element is a unit (that is, R× = R ∖ {0}) is called a division ring (or a skew-field). A commutative division ring is...

operations on Q {\displaystyle Q} , much like a division ring, but with some weaker conditions. All division rings, and thus all fields, are quasifields. A quasifield...

characterizes every simple Artinian ring as a ring of matrices over a division ring. This implies that a simple ring is left Artinian if and only if it...

reason is called a division ring). However, in other rings, division by nonzero elements may also pose problems. For example, the ring Z/6Z of integers...

Commutative ring: a ring in which the multiplication operation is commutative. Field: a commutative division ring (i.e. a commutative ring which contains...

mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra...

commutative) ring. The concept of a module also generalizes the notion of an abelian group, since the abelian groups are exactly the modules over the ring of integers...

dense subring of the ring of endomorphisms of a left vector space over a division ring. Another equivalent definition states that a ring is left primitive...

commutative, is called a division ring (or sometimes skew field). By Wedderburn's little theorem, any finite division ring is commutative, and hence...

fields and division rings. In a ring the elements by which division is always possible are called the units (for example, 1 and −1 in the ring of integers)...

mathematics, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if R and S are rings, then a ring homomorphism is...

3-sphere. The Hurwitz quaternions form an order (in the sense of ring theory) in the division ring of quaternions with rational components. It is in fact a maximal...

introduced by Nathan Jacobson (1944) for commutative fields and extended to division rings by Jacobson (1947), and Henri Cartan (1947) who credited the result...

mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates...

four-dimensional associative normed division algebra over the real numbers, and therefore a ring, also a division ring and a domain. It is a special case...

standard algebraic construction of systems satisfies these axioms. For a division ring D construct an (n + 1)-dimensional vector space over D (vector space...

and for any projective space defined arithmetically from a field or division ring; that includes any projective space of dimension greater than two or...

(x*)* = x for all x, y in A. This is also called an involutive ring, involutory ring, and ring with involution. The third axiom is implied by the second and...

R is isomorphic to a product of finitely many ni-by-ni matrix rings over division rings Di, for some integers ni, both of which are uniquely determined...