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Examples of commutative rings include the set of integers with their standard addition and multiplication, the set of polynomials with their addition and multiplication, the coordinate ring of an affine algebraic variety, and the ring of integers of a number field. Examples of noncommutative rings include the ring of n × n real square matrices ...
In mathematics, more specifically abstract algebra, a finite ring is a ring that has a finite number of elements. Every finite field is an example of a finite ring, and the additive part of every finite ring is an example of an abelian finite group, but the concept of finite rings in their own right has a more recent history.
Local algebra is the branch of commutative algebra that studies commutative local rings and their modules. In practice, a commutative local ring often arises as the result of the localization of a ring at a prime ideal. The concept of local rings was introduced by Wolfgang Krull in 1938 under the name Stellenringe. [1]
On the real line R, the family of sets consisting of the empty set and all finite unions of half-open intervals of the form (a, b], with a, b ∈ R is a ring in the measure-theoretic sense. If T is any transformation defined on a space, then the sets that are mapped into themselves by T are closed under both unions and intersections.
The ring Z / 3Z, or GF(3), has the elements 1, 0, and −1; its projective line has the four elements U[1, 0], U[1, 1], U[0, 1], U[1, −1] since both 1 and −1 are units. The homography group on this projective line has 12 elements, also described with matrices or as permutations. [1]: 31 For a finite field GF(q), the projective line is the ...
When is not a commutative ring, the idea of order is still important, but the phenomena are different. For example, the Hurwitz quaternions form a maximal order in the quaternions with rational co-ordinates; they are not the quaternions with integer coordinates in the most obvious sense. Maximal orders exist in general, but need not be unique ...
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).
The real numbers are an ordered ring which is also an ordered field. The integers, a subset of the real numbers, are an ordered ring that is not an ordered field. In abstract algebra, an ordered ring is a (usually commutative) ring R with a total order ≤ such that for all a, b, and c in R: [1] if a ≤ b then a + c ≤ b + c. if 0 ≤ a and 0 ...