Show that a ring R is simple left artinian⇐⇒ R

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2019 ALGEBRA II - MIDTERM EXAM

1. Let R and R⁰ be Morita similar. Show that R is primitive if and only if R⁰ is.

Also in any correspondence between ideals, J(R)corresponds to J(R⁰)_. 2. Show that a ring R is simple left artinian⇐⇒ R ∼= Mn(_∆) where∆ is a divi-

sion ring, n ∈_N.

3. Prove Frobenius’ theorem on division rings overR or Wedderburn’s theorem on finite division rings. You can use any method you know.

4. Let ρ, ρ⁰ be representations of G over an infinite field F. If ρ⊗_F K ∼= ρ⁰⊗_F K for an extension field K of F, show that ρ∼=ρ⁰. How about if|F| < _∞?

5. Let H ⊂ K ⊂ G be subgroups with G finite, σ and ρ are complex repre- sentations of H and G respectively. Using character calculations to prove (1) (σ^K)^G ∼=σ^G, (2) σ^G⊗ρ∼= (σ⊗ρH)^G, (3)(σ^G)^∗ ∼= (σ^∗)^G.

6. Show that the quaternion group Q8is not isomorphic to D4, but they have the same character table. Describe the rings F[Q₈]_{and F}[D₄]_{for F} =_{Q, C.}

7. Construct the character table for G = A5. (You get partial credits for doing the simpler case G =S4.)

8. Present an essential topic/theorem in modules, rings, or representations that you have well-prepared but not shown in the above problems.

Each problem is of 15 points (total 120 pts). Be sure to show your answers/computations/proofs in details. Time: pm 5:30 – 9:30, April 22, 2019 at AMB 101. A course by Chin-Lung Wang at NTU..

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