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Divisibility (ring theory) In mathematics, the notion of a divisor originally arose within the context of arithmetic of whole numbers. With the development of abstract rings, of which the integers are the archetype, the original notion of divisor found a natural extension. Divisibility is a useful concept for the analysis of the structure of ...
Division ring. In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring [1] in which every nonzero element a has a multiplicative inverse, that is, an element usually denoted a–1, such that a a–1 = a–1 a = 1.
t. e. In algebra, ring theory is the study of rings [1] — algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their representations, or, in different language, modules, special classes of rings (group rings ...
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 is not a field is the ring of quaternions. Any centralizer in a division ring is also a division ring. In particular, the center of a division ring is a field.
More generally, any root of unity in a ring R is a unit: if r n = 1, then r n−1 is a multiplicative inverse of r. In a nonzero ring, the element 0 is not a unit, so R × is not closed under addition. A 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 division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a nonzero ring [2] in which every nonzero element a has a multiplicative inverse, i.e., an element x with a · x = x · a = 1. Stated differently, a ring is a division ring if and only if its group of units is the set of all nonzero elements.
By Wedderburn's theorem, every finite division ring is commutative, and therefore a finite field. Another condition ensuring commutativity of a ring, due to Jacobson, is the following: for every element r of R there exists an integer n > 1 such that r n = r. [7] If, r 2 = r for every r, the ring is called Boolean ring. More general conditions ...
The ring of integers modulo a prime number has no nonzero zero divisors. Since every nonzero element is a unit, this ring is a finite field. More generally, a division ring has no nonzero zero divisors. A non-zero commutative ring whose only zero divisor is 0 is called an integral domain.
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