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

A matrix, also called a canonical box matrix, having zeros everywhere except along the diagonal and superdiagonal, with each element of the diagonal consisting of a single ...

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 ...

A complex vector space is a vector space whose field of scalars is the complex numbers. A linear transformation between complex vector spaces is given by a matrix with ...

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.

An invertible linear transformation T:V->W is a map between vector spaces V and W with an inverse map which is also a linear transformation. When T is given by matrix ...