Summary: Rings and Algebras Problem set #4. Oct. 6, 2010.
1. a) Let S R, and suppose RR is semisimple. Does it follow that SS is also semisimple?
b) Is it true that any ring S can be embedded into a semisimple ring R?
2. Determine which of the following abelian groups are semisimple:
Z, Q, Q/Z, Zn, Z2 Z3 Z5 · · · , Z2 × Z3 × Z5 × · · · , Z2 × Z2 × Z2 × · · · .
3. Let M be a semisimple module.
a) Show that if M is a direct sum of isomorphic simple modules, say M = S (such semisim-
ple modules are called homogeneous of type S), then any simple submodule of M is iso-
morphic to S.
b) Show that the decompisition of a semisimple module into a direct sum of simple modules
is not unique, however the submodules generated by isomorphic summands are uniquely
4. Let M, N be two R-modules. If f HomR(M, N), then there is an induced map ¯f
HomR(M/J(R)M, N/J(R)N. Show that if N is finitely generated then f is surjective if and
only if ¯f is surjective.
5. a) Show that the endomorphism ring of an artinian semisimple module is semisimple.
b) Prove the WedderburnArtin Theorem: A ring R is semisimple if and only if R is a ring
direct sum of finitely many ideals, each of which is the full matrix rings over a division
6. Let M be a finite dimensional complex vector space which is also a module over C[x]. Let