Then A is an Artinian ring. Commutative case As A is Artinian, if it is commutative, then it is a finite product of Artinian local rings whose residue fields are algebras over the base field k. Now, a reduced Artinian local ring is a field and thus the following are equivalent is separable. is reduced, where is some algebraic closure of k. for some n. is the number of -algebra homomorphisms . Noncommutative case Since a simple Artinian ring is a (full) matrix ring over a division ring, if A is a simple algebra, then A is a (full) matrix algebra over a division algebra D over k; i.e., . More generally, if A is a semisimple algebra, then it is a finite product of matrix algebras (over various division k-algebras), the fact known as the Artin–Wedderburn theorem. The fact that A is Artinian simplifies the notion of a Jacobson radical; for an Artinian ring, the Jacobson radical of A is the intersection of all (two-sided) maximal ideals (in contrast, in general, a Jacobson radical is the intersection of all left maximal ideals or the intersection of all right maximal ideals.) The Wedderburn principal theorem states: for a finite-dimensional algebra A with a nilpotent ideal I, if the projective dimension of as an -module is at most one, then the natural surjection splits; i.e., contains a subalgebra such that is an isomorphism. Taking I to be the Jacobson radical, the theorem says in particular that the Jacobson radical is complemented by a semisimple algebra. The theorem is an analog of Levi's theorem for Lie algebras. Lattices and orders Let R be a Noetherian integral domain with field of fractions K (for example, they can be ). A lattice L in a finite-dimensional K-vector space V is a finitely generated R-submodule of V that spans V; in other words, . Let be a finite-dimensional K-algebra. An order in is an R-subalgebra that is a lattice. In general, there are a lot fewer orders than lattices; e.g., is a lattice in but not an order (since it is not an algebra). A maximal order is an order that is maximal among all the orders. Related concepts Coalgebras An associative algebra over K is given by a K-vector space A endowed with a bilinear map A × A → A having two inputs (multiplicator and multiplicand) and one output (product), as well as