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A module taking its coefficients in a ring R is called a module over R or R-module.

The vector r from the origin to the current position. It is also called the position vector. The derivative of r satisfies ...

A module over a ring that is contained in and has the same addition as another module over the same ring.

Let E be a Euclidean space, (beta,alpha) be the dot product, and denote the reflection in the hyperplane P_alpha={beta in E|(beta,alpha)=0} by ...

Given a module M over a unit ring R, the set End_R(M) of its module endomorphisms is a ring with respect to the addition of maps, (f+g)(x)=f(x)+g(x), for all x in M, and the ...

A ring equipped with a filtration.

A filtration of ideals of a commutative unit ring R is a sequence of ideals ... subset= I_2 subset= I_1 subset= I_0=R, such that I_iI_j subset= I_(i+j) for all indices i,j. ...

A semigroup with a noncommutative product in which no product can ever be expressed more simply in terms of other elements.

The set of sums sum_(x)a_xx ranging over a multiplicative group and a_i are elements of a field with all but a finite number of a_i=0. Group rings are graded algebras.

The commutation operation [a,b]=ab-ba corresponding to the Lie product.