Abstract Algebra Answers

Let us call a ring A a matrix ring if A ∼ Mm(R) for some integer m ≥ 2and some ring R. True or False: “A homomorphic image of a matrix ring is also a matrix ring”?

Let R, S be rings such that Mm(R) ∼= Mn(S). Does this imply that m = n and R ∼ S?

R be a simple ring that is finite-dimensional over its center k, show that R is isomorphic to a matrix
algebra over its center k iff R has a nonzero left ideal A with (dimkA)2 ≤ dimkR

Let R be a simple ring that is finite-dimensional over its center k. Let M be a finitely generated left R-module and let E = End(RM). Show that (dimkM)2 = (dimkR)(dimkE).

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 • •