I feel silly asking this, but what is the importance of the Artin-Wedderburn theorem in Algebra?
In A First Course in Noncommutative Rings, T.Y. Lam calls it "the cornerstone of non-commutative ring theory", and he goes on to list some very remarkable properties of modules over semisimple rings, some of which I list below :
- all modules are projective and injective,
- the ring itself is isomorphic to a product of matrix rings over division rings, and both these division rings and the size of these matrices are well defined (this is of course the Artin-Wedderburn theorem),
- group rings are semisimple (except for some special cases).
As someone who knows little to nothing about algebra, even I understand that these are terrific properties to have. However, when do semisimple rings ever occur? Do they only occur as group rings (not that this isn't very important)? Also, is it possible to know the division rings $D_i$ that appear in the theorem (apart from their characterisation as the endomporhism rings of the simple submodules of $R$) and the sizes $n_i$ of the matrices, say if
$$ R\simeq\prod_{i=1}^r\mathrm{Mat}_{n_i}(D_i)~? $$
Best Answer
Let $R$ be an arbitrary ring. The quotient $R/J(R)$, where $J(R)$ is the Jacobson radical, is the part of $R$ that acts nontrivially on the simple modules of $R$; said another way, it is the universal semiprimitive quotient of $R$. If $R$ is Artinian, then $R/J(R)$ is Artinian semiprimitive, which turns out to be equivalent to semisimple, so we can apply Artin-Wedderburn to $R/J(R)$ and learn something important about our original ring $R$ (namely everything we can learn from looking at simple modules).