Ring theory

In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts.

In mathematics, the symmetric algebra S(V) (also denoted Sym(V)) on a vector space V over a field K is the free commutative unital associative algebra over K containing V.

In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative.

In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic p, an important class which includes finite fields.

In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension.

An integer (from the Latin integer meaning "whole") is a number that can be written without a fractional component.

In mathematics, and specifically in abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero.

In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.

In number theory, quadratic integers are a generalization of the integers to quadratic fields.

In mathematics, an invertible element or a unit in a (unital) ring R is any element u that has an inverse element in the multiplicative monoid of R, i.

A geometric algebra (GA) is a Clifford algebra of a vector space over the field of real numbers endowed with a quadratic form.

In commutative algebra, an element b of a commutative ring B is said to be integral over A, a subring of B, if there are n ≥ 1 and such that That is to say, b is a root of a monic polynomial over A.

In algebra, a Hilbert ring or a Jacobson ring is a ring such that every prime ideal is an intersection of primitive ideals.

In abstract algebra, in particular in the theory of nondegenerate quadratic forms on vector spaces, the structures of finite-dimensional real and complex Clifford algebras have been completely classified.

In algebra, a torsion-free module is a module over a ring such that 0 is the only element annihilated by a regular element (non zero-divisor) of the ring.