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For example, (Z/2Z, +) is a quotient of (Z/4Z, +), using the morphism consisting of taking the remainder modulo 2 of an integer. A semigroup T divides a semigroup S, denoted T ≼ S if T is a quotient of a subsemigroup S. In particular, subsemigroups of S divides T, while it is not necessarily the case that there are a quotient of S.
A division algorithm is an algorithm which, given two integers N and D (respectively the numerator and the denominator), computes their quotient and/or remainder, the result of Euclidean division. Some are applied by hand, while others are employed by digital circuit designs and software. Division algorithms fall into two main categories: slow ...
Cultural intelligence. Cultural intelligence or cultural quotient ( CQ ), refers to an individual's capability to function effectively in culturally diverse settings. The concept was introduced by London Business School professor P. Christopher Earley and Nanyang Business School professor Soon Ang in 2003. [1] [2]
Reduced ring. In ring theory, a branch of mathematics, a ring is called a reduced ring if it has no non-zero nilpotent elements. Equivalently, a ring is reduced if it has no non-zero elements with square zero, that is, x2 = 0 implies x = 0. A commutative algebra over a commutative ring is called a reduced algebra if its underlying ring is reduced.
In abstract algebra, a valuation ring is an integral domain D such that for every non-zero element x of its field of fractions F, at least one of x or x −1 belongs to D.. Given a field F, if D is a subring of F such that either x or x −1 belongs to D for every nonzero x in F, then D is said to be a valuation ring for the field F or a place of F.
Congruence relation. In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in the sense that algebraic operations done with equivalent elements will yield equivalent elements. [1]
The quotient map R/I → R/J has a non-zero kernel which is not equal to R/I, and therefore R/I is not simple. Every simple R-module is isomorphic to a quotient R/m where m is a maximal right ideal of R. By the above paragraph, any quotient R/m is a simple module. Conversely, suppose that M is a simple R-module.
with -basis {,,,,,} as in the previous example. Therefore the Galois group = (/) has six elements, determined by all permutations of the three roots of : =, =, =. Since there are only 3! = 6 such permutations, G must be isomorphic to the symmetric group of all permutations of three objects.