Summary: Rings and Algebras Problem set #4: Solutions Sept. 6, 2011.
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?
Solution. a) No, take for example Z Q. b) We know that any semisimple ring is artinian (and noetherian) since the identity
element is contained in a finite direct sum of simple modules, hence RR is a finite direct sum. Now, if S = R1 × R2 × ˇ ˇ ˇ is the
cartesian product of infinitely many (nontrivial) rings, then the identity elements of the components will give an inifinite set e1, e2, . . .
of orthogonal idempotents in S and hence also in any ring R containing S. Then Re1 R(e1 + e2) ˇ ˇ ˇ R(e1 + e2 + ˇ ˇ ˇ + en) ˇ ˇ ˇ
is a strictly increasing sequence of left ideals in R, hence R is not noetherian. Thus S cannot be embedded into a semisimple ring.
2. Determine which of the following abelian groups are semisimple:
Z, Q, Q/Z, Zn, Z2 Z3 Z5 ˇ ˇ ˇ , Z2 × Z3 × Z5 × ˇ ˇ ˇ , Z2 × Z2 × Z2 × ˇ ˇ ˇ .
Solution. Observe that any submodule of a semisimple module is semisimple. (This does not contradict the fact seen in the previous
problem that semisimple rings can have non-semisimle subrings.) For example if M is semisimple and N M then N is also a direct
summand, hence a homomorphic image of M. The images of the simple submodules of M will be simple and will generate N, hence
N is also semisimple. Since ZZ is not semisimple it does not have any simple submodules semisimple abelian groups cannot have
torsion-free elements. Thus we get that Z, Q and Z2 × Z3 × Z5 × ˇ ˇ ˇ are not semisimple. Z2 Z3 Z5 ˇ ˇ ˇ is almost by definition
semisimple, and similarly, Z2 × Z2 × Z2 × ˇ ˇ ˇ is semisimple since each of its nonzero elements is contained in a simple abelian group
Z2. The group Q/Z contains submodules isomorphic to Zp which are not semisimple (they are not complemented) hence it is not
semisimple. Finally, Zn is semisimple if and only if n is squarefree.
3. Let M be a semisimple module.
a) Show that if M is a direct sum of isomorphic simple modules, say M = S (such semisimple modules are