vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and the algebraic operations in terms of matrix addition and matrix multiplication, which is non-commutative. The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is the representation theory of groups, in which elements of a group are represented by invertible matrices in such a way that the group operation is matrix multiplication. Some relevant theorems General Isomorphism theorems for rings Nakayama's lemma Structure theorems The Artin–Wedderburn theorem determines the structure of semisimple rings The Jacobson density theorem determines the structure of primitive rings Goldie's theorem determines the structure of semiprime Goldie rings The Zariski–Samuel theorem determines the structure of a commutative principal ideal ring The Hopkins–Levitzki theorem gives necessary and sufficient conditions for a Noetherian ring to be an Artinian ring Morita theory consists of theorems determining when two rings have "equivalent" module categories Cartan–Brauer–Hua theorem gives insight on the structure of division rings Wedderburn's little theorem states that finite domains are fields Other The Skolem–Noether theorem characterizes the automorphisms of simple rings Structures and invariants of rings Dimension of a commutative ring In this section, R denotes a commutative ring. The Krull dimension of R is the supremum of the lengths n of all the chains of prime ideals . It turns out that the polynomial ring over a field k has dimension n. The fundamental theorem of dimension theory states that the following numbers coincide for a noetherian local ring : The Krull dimension of R. The minimum number of the generators of the -primary ideals. The dimension of the graded ring (equivalently, 1 plus the degree of its Hilbert polynomial). A commutative ring R is said to be catenary if for every pair of prime ideals , there exists a finite chain of prime ideals that is maximal in the sense that it is impossible to insert an additional prime ideal between two ideals in the chain, and all such maximal chains between and have the same length. Practically all noetherian rings that appear in applications are catenary. Ratliff proved that a noetherian local integral domain R is catenary if and only if for every prime ideal , where is the height of . If