of the Jordan normal form. Non-diagonal Jordan forms occur when the characteristic divides the order of the group. Ring theory interpretation Given a field K and a finite group G, the group algebra K[G] (which is the K-vector space with K-basis consisting of the elements of G, endowed with algebra multiplication by extending the multiplication of G by linearity) is an Artinian ring. When the order of G is divisible by the characteristic of K, the group algebra is not semisimple, hence has non-zero Jacobson radical. In that case, there are finite-dimensional modules for the group algebra that are not projective modules. By contrast, in the characteristic 0 case every irreducible representation is a direct summand of the regular representation, hence is projective. Brauer characters Modular representation theory was developed by Richard Brauer from about 1940 onwards to study in greater depth the relationships between the characteristic p representation theory, ordinary character theory and structure of G, especially as the latter relates to the embedding of, and relationships between, its p-subgroups. Such results can be applied in group theory to problems not directly phrased in terms of representations. Brauer introduced the notion now known as the Brauer character. When K is algebraically closed of positive characteristic p, there is a bijection between roots of unity in K and complex roots of unity of order prime to p. Once a choice of such a bijection is fixed, the Brauer character of a representation assigns to each group element of order coprime to p the sum of complex roots of unity corresponding to the eigenvalues (including multiplicities) of that element in the given representation. The Brauer character of a representation determines its composition factors but not, in general, its equivalence type. The irreducible Brauer characters are those afforded by the simple modules. These are integral (though not necessarily non-negative) combinations of the restrictions to elements of order coprime to p of the ordinary irreducible characters. Conversely, the restriction to the elements of order coprime to p of each ordinary irreducible character is uniquely expressible as a non-negative integer combination of irreducible Brauer characters. Reduction (mod p) In the theory initially developed by Brauer, the link between ordinary representation theory and modular representation theory is best exemplified by considering the group algebra of the group G over a complete discrete valuation ring R with residue field K of positive characteristic p and field