# Division Ring

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## division ring

[di′vizh·ən ‚riŋ]
(mathematics)
A ring in which the set of nonzero elements form a group under multiplication.
More generally, a nonassociative ring with nonzero elements in which, for any two elements a and b, there are elements x and y such that ax = b and ya = b.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## Division Ring

a set of elements for which operations of addition, subtraction, multiplication, and division are defined that have the usual properties of the operations on numbers, except that the operation of multiplication need not be commutative. The set of quaternions is an example of a division ring. If multiplication of elements of a division ring is commutative, the division ring is a field.

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Let [GAMMA] = (P, L) be a point-line geometry, let M be the set of point-line flags of [GAMMA], and let D be a division ring. A point-line presheaf F = ([{[F.sub.p]}.sub.p], [{[F.sub.L]}.sub.L], [{([[phi].sub.pL]}.sub.pL]) on [GAMMA] over D, or just a D-presheaf on [GAMMA], is a set of 1-dimensional left vector spaces {[F.sub.p]|p [member of] P} over D, a set of 2-dimensional left vector spaces {[F.sub.L]|L [member of] L} over D, and a set of injective D-linear connecting maps {[[phi].sub.pL] : [F.sub.p] [right arrow] [F.sub.L]|(p,L), [member of] M}.
To contact the Huddersfield division ring 01484 425472 or 01484 533462.
By Martindale's theorem (18), R is then a primitive ring having nonzero socle H with C as the associated division ring. Hence by Jacobson's theorem [13, p.75] R is isomorphic to a dense ring of linear transformations of some vector space V over C, and H consists of the linear transformations in R of finite rank.
Proposition 2.6.() Any finite domain is a division ring.
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Hua's fundamental theorem of geometry of matrices describes the general form of bijective maps on the space of all m X n matrices over a division ring D that preserve adjacency in both directions, explains Semrl.
Therefore H cannot contain two minimal orthogonal idempotent elements and so H = D, for a suitable division ring D finite dimensional over its center.

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