In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist...

an algebra may or may not be associative, leading to the notions of associative algebras and non-associative algebras. Given an integer n, the ring of...

abstract algebra, a *-algebra (or involutive algebra; read as "star-algebra") is a mathematical structure consisting of two involutive rings R and A,...

especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials...

In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite...

In mathematics, an associative algebra A over a commutative ring (often a field) K is a ring A together with a ring homomorphism from K into the center...

for noncommutative rings. An algebra is unital or unitary if it has an identity element e with ex = x = xe for all x in the algebra. For example, the octonions...

Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic...

especially in the area of abstract algebra known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring since its elements may be described...

a De Morgan algebra and a Kleene algebra (with involution). Every Boolean algebra gives rise to a Boolean ring, and vice versa, with ring multiplication...

Equivalently, a noncommutative ring is a ring that is not a commutative ring. Noncommutative algebra is the part of ring theory devoted to study of properties...

In abstract algebra, a matrix ring is a set of matrices with entries in a ring R that form a ring under matrix addition and matrix multiplication. The...

of a ring R is a subring of R. This article also deals with centralizers and normalizers in a Lie algebra. The idealizer in a semigroup or ring is another...

and two-sided ideals for rings. Kernels allow defining quotient objects (also called quotient algebras in universal algebra, and cokernels in category...

specifically in abstract algebra, a rng (or non-unital ring or pseudo-ring) is an algebraic structure satisfying the same properties as a ring, but without assuming...

need only be a ring, so the module concept represents a significant generalization. In commutative algebra, both ideals and quotient rings are modules,...

commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily...

elements. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term abstract algebra was coined...

in algebraic terms. The definition of a Lie algebra over a field extends to define a Lie algebra over any commutative ring R. Namely, a Lie algebra g {\displaystyle...

In mathematics, the ring of integers of an algebraic number field K {\displaystyle K} is the ring of all algebraic integers contained in K {\displaystyle...

equivalent to a purely algebraic definition as an algebra of symmetries. Two basic examples of von Neumann algebras are as follows: The ring L ∞ ( R ) {\displaystyle...

commutative ring. The collection of all structures of a given type (same operations and same laws) is called a variety in universal algebra; this term...

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

In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R i {\displaystyle...

modulo 2. Every Boolean ring gives rise to a Boolean algebra, with ring multiplication corresponding to conjunction or meet ∧, and ring addition to exclusive...

structures, such as groups, rings, and fields, based on the number of operations they use and the laws they follow. Universal algebra and category theory provide...

In algebra, the center of a ring R is the subring consisting of the elements x such that xy = yx for all elements y in R. It is a commutative ring and...

ideal is also defined for non-associative rings (often without the multiplicative identity) such as a Lie algebra. (For the sake of brevity, some results...

homomorphism R → End(M). A unital algebra homomorphism between unital associative algebras over a commutative ring R is a ring homomorphism that is also R-linear...

In mathematical analysis and in probability theory, a σ-algebra ("sigma algebra"; also σ-field, where the σ comes from the German "Summe") on a set X...