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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 ...

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

The multiplication operation corresponding to the Lie bracket.

A sum of the elements from some set with constant coefficients placed in front of each. For example, a linear combination of the vectors x, y, and z is given by ax+by+cz, ...

The kernel of a linear transformation T:V-->W between vector spaces is its null space.

A noncommutative ring R is a ring in which the law of multiplicative commutativity is not satisfied, i.e., a·b!=b·a for any two elements a,b in R. In such a case, the ...

In a space E equipped with a symmetric, differential k-form, or Hermitian form, the orthogonal sum is the direct sum of two subspaces V and W, which are mutually orthogonal. ...

A linear transformation x_1^' = a_(11)x_1+a_(12)x_2+a_(13)x_3 (1) x_2^' = a_(21)x_1+a_(22)x_2+a_(23)x_3 (2) x_3^' = a_(31)x_1+a_(32)x_2+a_(33)x_3, (3) is said to be an ...