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.

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.

References in periodicals archive ?
This will resolve the spatial organization of the fascinating patterns of Min proteins and chromatin that dictate the localization of the division ring.
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.
Therefore H cannot contain two minimal orthogonal idempotent elements and so H = D, for a suitable division ring D finite dimensional over its center.
Other topics include Temperley-Lieb and non-crossing partition planar algebras, Wedderburn polynomials over division rings, generic irreducibles of the Brauer algebras, and Hopf monads on categories.
Unlike Turiya, Kitchen & Home and Colonial Kitchens & Garden -- three unprofitable businesses that Hanover shut down in the fourth quarter -- the Gump's division rings up sales of about $75 million annually, accounting for slightly more than 10 percent of Hanover's total revenues, which came in around $600 million for 2000.
Innovative food manufacturer's Grocery and Diversified Products division rings up operating profits of $918.
These modules play important roles in various areas of algebra, primarily commutative algebra, including rings of p-adic integers and certain power series rings over division rings.

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