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,...

In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more...

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...

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...

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...

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...

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...

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

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 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...

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

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

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...

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...

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

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

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...

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...

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

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...

Therefore, the symmetric algebra over V can be viewed as a "coordinate free" polynomial ring over V. The symmetric algebra S(V) can be built as the quotient...

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, 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...

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...

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...

In algebra, a unit or invertible element of a ring is an invertible element for the multiplication of the ring. That is, an element u of a ring R is a...

endomorphism ring consequently encodes several internal properties of the object. As the endomorphism ring is often an algebra over some ring R, this may...

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...