2019 ALGEBRA II - QUIZ I
1. Let R and S be two local rings such that Mm(R) ∼= Mn(S). Show that then m=n and R ∼=S. Give a counterexample with non-local R.
2. Let F be a field. Use Zorn’s Lemma to show that any vector space V over F has a basis and any two basis have the same cardinality.
3. Prove a generalization ofZ/(m) ⊗Z/(n) ∼=Z/(gcd(m, n))to the case when Z is replaced by a p.i.d. R. Then determine the structure of M1⊗R M2 of finitely generated modules M1, M2over R.
4. Let R, S be rings. Let P = RP be f.g. projective. M = RMS, N = SN. Show that there is a group isomorphism
η : homR(P, M) ⊗S N →homR(P, M⊗S N).
Give a counterexample with R=S =Z and P=Z/(n), n ∈N.
5. Given a Morita context(R, R0,R0PR,RPR00, τ, µ)with τ : P0⊗R0 P → R and µ : P⊗RP0 →R0being surjective. Prove part of Morita I: (1) PRis a progenerator, (2) τ is an isomorphism, (3) P0 ∼=P∗.
Show your answers/computations/proofs in details. Date: pm 2:50 – 3:50, March 15, 2019 at AMB 101. A course by Chin-Lung Wang at NTU..
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