Ring Theory Interpretation
Given a field K and a finite group G, the group algebra K (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.
Read more about this topic: Modular Representation Theory
Famous quotes containing the words ring and/or theory:
“Look how my ring encompasseth thy finger;
Even so thy breast encloseth my poor heart.
Wear both of them, for both of them are thine.”
—William Shakespeare (1564–1616)
“The weakness of the man who, when his theory works out into a flagrant contradiction of the facts, concludes “So much the worse for the facts: let them be altered,” instead of “So much the worse for my theory.””
—George Bernard Shaw (1856–1950)