Summary: Rings and Algebras Problem set #4. Oct. 6, 2010.
1. a) Let S # R, and suppose RR is semisimple. Does it follow that S S 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, Z n , Z 2 # Z 3 # Z 5 # · · · , Z 2 × Z 3 × Z 5 × · · · , Z 2 × Z 2 × Z 2 × · · · .
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 Rmodules. If f # HomR (M, N ), then there is an induced map •
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 Wedderburn--Artin 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