Emmy Noether introduced the notion of a representation space— a vector space upon which the elements of the algebra operate as linear transformations, the composition of the linear transformations reflecting the multiplication in the algebra. By doing so she enables us to use our geometric intuition. Her point of view stresses the essential fact about a simple algebra, namely, that it has only one type of irreducible space and that it is faithfully represented by its operation on this space. Wedderburn's statement that the simple algebra is a total matrix algebra over a quasifield is now more understandable. It simply means that all transformations of this space which are linear with respect to a certain quasifield are produced by the algebra. This treatment of algebras may be found in van der Waerden's Moderne Algebra. Recently it has been discovered that this last described treatment of simple algebras is capable of generalization to a far wider class of rings.

What It Means

This is a placeholder explanation generated by the low-overhead model. It interprets "Emmy Noether introduced the notion of a representation space— a vector space upon which the elements of the algebra operate as linear transformations, the composition of the linear transformations reflecting the multiplication in the algebra. By doing so she enables us to use our geometric intuition. Her point of view stresses the essential fact about a simple algebra, namely, that it has only one type of irreducible space and that it is faithfully represented by its operation on this space. Wedderburn's statement that the simple algebra is a total matrix algebra over a quasifield is now more understandable. It simply means that all transformations of this space which are linear with respect to a certain quasifield are produced by the algebra. This treatment of algebras may be found in van der Waerden's Moderne Algebra. Recently it has been discovered that this last described treatment of simple algebras is capable of generalization to a far wider class of rings." in the context of "Emil Artin, (1950), "The influence of J. H. M. Wedderburn on the development of modern algebra", Bulletin of the American Mathematical Society 56 no: 1, pages: 65–72 (quote from p. 67)" to mean that wisdom is timeless.