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? 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 Collections: Mathematics