Abstract Algebra Answers

Let M be a left R-module and E = End(RM), if E is a semisimple E-module, is M necessarily a semisimple R-module?

Let R be a domain. Show that if Mn(R) is semisimple, then R is a division ring.

Show that if R is semisimple, so is Mn(R).

Show that, if M is a simple module over a ring R, then as an abelian group, M is isomorphic to a direct sum of copies of Q, or a direct sum of copies of Zp for some prime p.

Let V be a left R-module with elements e1, e2, . . . such that, for any n, there exists r ∈ R such ren, ren+1, . . . are almost all 0, but not all 0. Show that P := V × V ו • • is not a semisimple R-module.

Let V be a left R-module with elements e1, e2, . . . such that, for any n, there exists r ∈ R such ren, ren+1, . . . are almost all 0, but not all 0. Show that S := V ⊕ V ⊕• • • is not a direct summand of P := V × V ו • •

a frog lies at the bottom of a well whose depth is 30m. It starts to climb, rising up 2/5 every day and falling back 1/5 every night. How long will it take him to climb out of the well?

Does this hold over an arbitrary ring R: any direct product of copies of V is semisimple?

Let U, V be semisimple modules over a commutative ring. Is HomR(U, V ) also semisimple?

Let U, V be modules over a commutative ring R. If U or V is semisimple, show that U ⊗R V is also semisimple.