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...

commutative ring Divisibility (ring theory): nilpotent element, (ex. dual numbers) Ideals and modules: Radical of an ideal, Morita equivalence Ring homomorphisms:...

Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a...

In mathematics, specifically in the field of group theory, a divisible group is an abelian group in which every element can, in some sense, be divided...

on its properties. Commutative algebra, the theory of commutative rings, is a major branch of ring theory. Its development has been greatly influenced...

generalizations of the ring of integers and provide a natural setting for studying divisibility. In an integral domain, every nonzero element a has the cancellation...

example where this is not the case. Another example is given by the divisibility (or "is-a-factor-of") relation |. For two natural numbers n and m, we...

collection of divisibility events associated to distinct primes is mutually independent. For example, in the case of two events, a number is divisible by primes...

units −1 and 1 and prime numbers have no non-trivial divisors. There are divisibility rules that allow one to recognize certain divisors of a number from the...

a ring homomorphism. In this case, f is called a ring isomorphism, and the rings R and S are called isomorphic. From the standpoint of ring theory, isomorphic...

defined by the divisibility by m and because −1 is a unit in the ring of integers, a number is divisible by −m exactly if it is divisible by m. This means...

Algebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic...

quantity. Elementary number theory studies divisibility rules in order to quickly identify if a given integer is divisible by a fixed divisor. For instance...

In mathematics, and more specifically in ring theory, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the...

supplements introducing the notion of an ideal, fundamental to ring theory. (The word "Ring", introduced later by Hilbert, does not appear in Dedekind's...

name of what is now known as Cavalieri's principle Absence of divisibility (ring theory) Indivisible (2016 film), a 2016 Italian film Indivisible (2018...

(d)cpo Bounded complete Complete lattice Knaster–Tarski theorem Infinite divisibility Heyting algebra Relatively complemented lattice Complete Heyting algebra...

In algebra, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those...

classical formula applying to the set of the natural numbers ordered by divisibility: see incidence algebra. The classic version states that if g and f are...

generalized to sequences with values in any ring where the concept of divisibility is defined. A strong divisibility sequence is an integer sequence ( a n )...

quotient rings Z / n Z {\displaystyle \mathbb {Z} /n\mathbb {Z} } runs through all natural numbers n {\displaystyle n} , partially ordered by divisibility. By...

integral quadratic forms. The study of rings of quadratic integers is basic for many questions of algebraic number theory. Medieval Indian mathematicians had...

endomorphism ring is simply the ring of formal power series. If G is a finite group and k a field with characteristic 0, then one shows in the theory of group...

mathematics, the adele ring of a global field (also adelic ring, ring of adeles or ring of adèles) is a central object of class field theory, a branch of algebraic...

inclusion (i.e. the elements in D are, up to units, totally ordered by divisibility.) There is a totally ordered abelian group Γ (called the value group)...

.} are partially ordered by divisibility, then 1 is the smallest and 0 is the largest. Then the characteristic of a ring is the smallest value of n for...

extending the multiplication of G by linearity) is an Artinian ring. When the order of G is divisible by the characteristic of K, the group algebra is not semisimple...

_{i=0}^{\infty }{\frac {x^{i}n_{i}}{i!}}.} In number theory, the most salient property of factorials is the divisibility of n ! {\displaystyle n!} by all positive...

discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential...

Another example is given by the natural numbers, partially ordered by divisibility, for which the supremum is the least common multiple and the infimum...