Editing Modular representation theory (section)

== Ring theory interpretation ==

Given a field ''K'' and a finite group ''G'', the [[group ring|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 algebraic group|semisimple]], hence has non-zero [[Jacobson radical]]. In that case, there are finite-dimensional modules for the group algebra that are not [[projective module]]s. By contrast, in the characteristic 0 case every [[irreducible representation]] is a [[direct summand]] of the [[regular representation]], hence is projective.