Ring theory

thumb|upright=1.25|The integers arranged on a number line

algebraic structure that has compatible structures of an abelian group and a monoid, in particular having multiplicative identity

branch of abstract algebra in mathematics

algebraic structure

commutative ring with no zero divisors other than zero

ring in which division is possible

In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that x^n=0. The smallest such n is called the index of nilpotency or the degree of nilpotency of x.

in a field or a ring, the smallest positive integer, if any, such that the sum of n ones equals 0; zero otherwise

ring element such that can be multiplied by a non-zero element to equal 0

ring of polynomials (with one or several variables) with coefficients in a given ring

Commutative ring with an Euclidean division

Algebraic structure

ring whose ideals satisfy the ascending chain condition

integral domain where every nonzero element is uniquely expressible as a product of prime elements

construction in abstract algebra

generalization of a polynomial, where the number of terms is allowed to be infinite, defined algebraically without consideration of convergence (so that e.g. evaluation is not always defined)

algebraic structure generated by a vector space with a quadratic form and a unital associative algebra structure

in mathematics, an invertible element or a unit in a ring R

algebraic construction

proper ideal such that the only ideal it is properly contained in is the ring itself

In abstract algebra, a semiring is an algebraic structure. Semirings are a generalization of rings, dropping the requirement that each element must have an additive inverse. At the same time, semirings are a generalization of bounded distributive lattices.

an element which, when added to any element x in the set, yields x

mapping between two rings which respects the structure

ring with unique one-sided maximal ideal

ring that satisfies the descending chain condition on ideals

nonzero, non-unit element p in a commutative ring R such that, whenever p divides ab for some a and b in R, then p divides a or p divides b (or both)

subgroup of a group G that each leaves invariant each element of a given subset of a G-set

In mathematics, a subring of a ring is a subset of that is itself a ring when binary operations of addition and multiplication on R are restricted to the subset, and that shares the same multiplicative identity as .

a natural number whose square "ends" in the same digits as the number itself

non-zero non-unit element in an integral domain that is not a product of two non-units

In mathematics, and more specifically in abstract algebra, a *-algebra (or involutive algebra; read as "star-algebra") is a mathematical structure consisting of two involutive rings and , where is commutative and has the structure of an associative algebra over . Involutive algebras generalize the idea of a number system equipped with conjugation, for example the complex numbers and complex conjugation, matrices over the complex numbers and conjugate transpose, and linear operators over a Hilbert space and Hermitian adjoints. However, it may happen that an algebra admits no involution.

radical of a ring as a module over itself; the intersection of all maximal right (or equivalently left) ideals of the ring; the sum of all superfluous right (or equiv. left) ideals

rational number whose denominator is a power of two

mathematical concept

mathematical element

graded module, where the grading has the structure of a monoid, in which ring multiplication respects the grading

Noetherian local commutative ring, the minimal number of generators of whose maximal ideal equals its Krull dimension

ring where ring operations are continuous

a field that is either of characteristic 0, or of positive characteristic p such that every element admits a p-th root

unique ring consisting of one element

Hopf algebra associated to a Lie algebra

direct sum of irreducible modules

nonzero ring that has no two-sided ideal besides the zero ideal and itself

module over a ring whose multiplication is bilinear

algebraic structure similar to ring but not necessarily having a multiplicative identity

algebra over a field with only invertible elements and zero

In abstract algebra, the biquaternions are the numbers , where , and are complex numbers, or variants thereof, and the elements of multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions corresponding to complex numbers and the variations thereof: Biquaternions when the coefficients are complex numbers. Split-biquaternions when the coefficients are split-complex numbers. Dual quaternions when the coefficients are dual numbers.

construction of a ring of fractions, in commutative algebra

unital ring with no zero divisors other than 0; noncommutative generalization of integral domains

polynomial with finitely many terms of the form axⁿ where n ∈ ℤ

algebra of all possible symmetric tensors over a vector space or ring module

ring of all matrices of fixed size

abstract algebra concept

integral domain in which the sum of two principal ideals is again a principal ideal

ring whose only nilpotent element is 0

differential algebra

element x of a ring such that x² = x

free module and at the same time a ring

Concept in algebra

endomorphism algebra of an abelian group