Difference between revisions of "Division ring"
From Encyclopedia of Mathematics
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| − | A (not necessarily associative) [[ | + | A ring with division is a (not necessarily associative) [[ring]] in which the equations |
| − | + | $$a \cdot x = b \quad;\quad\quad y \cdot a = b $$ | |
| − | are solvable for any two elements | + | |
| + | are solvable for any two elements $a$ and $b$, where $a \ne 0$. If the solutions of these equations are uniquely determined, then the ring is called a quasi-division ring. In contrast to an arbitrary division ring, a quasi-division ring cannot have divisors of zero (cf. [[Zero divisor|Zero divisor]]); the non-zero elements of a quasi-division ring form a [[Quasi-group|quasi-group]] with respect to multiplication. Each (not necessarily associative) ring without divisors of zero can be imbedded in a quasi-division ring. An associative division ring is an (associative) [[Skew-field|skew-field]]. See also [[Division algebra|Division algebra]]. | ||
====References==== | ====References==== | ||
Latest revision as of 17:50, 28 December 2013
A ring with division is a (not necessarily associative) ring in which the equations
$$a \cdot x = b \quad;\quad\quad y \cdot a = b $$
are solvable for any two elements $a$ and $b$, where $a \ne 0$. If the solutions of these equations are uniquely determined, then the ring is called a quasi-division ring. In contrast to an arbitrary division ring, a quasi-division ring cannot have divisors of zero (cf. Zero divisor); the non-zero elements of a quasi-division ring form a quasi-group with respect to multiplication. Each (not necessarily associative) ring without divisors of zero can be imbedded in a quasi-division ring. An associative division ring is an (associative) skew-field. See also Division algebra.
References
| [1] | A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian) |
Comments
References
| [a1] | N. Jacobson, "The theory of rings" , Amer. Math. Soc. (1943) |
How to Cite This Entry:
Division ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Division_ring&oldid=17857
Division ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Division_ring&oldid=17857
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article