10 Questions on Isomorphism in Abstract Algebra - Final Examination | MATH 647, Exams of Abstract Algebra

Download 10 Questions on Isomorphism in Abstract Algebra - Final Examination | MATH 647 and more Exams Abstract Algebra in PDF only on Docsity! 1 Math 648 Final Answer as many questions as you can! Make sure you state clearly any theorems from class that you use. Part I. Definitions. 1. For any (S, R)-bimodule P , (P⊗R?,HomS(P, ?)) is an adjoint pair of functors. Explain carefully what this sentence means. Part II. True or False. Justify your answers briefly. 1. Q⊗Z R ∼= R as an abelian group. 2. If M is a free R-module and X ⊆M is a minimal spanning set, then X is a basis for M . 3. If G is a finite abelian group and Z(p) is the localization of Z at the prime ideal (p), then Z(p) ⊗Z G is a p-group. 4. Any R-module of finite length can be decomposed into a direct sum of cyclic submodules. 5. For any ring R, the map f $→ f(1R) is a ring isomorphism EndR(RR) ∼→ R. Part III. Longer problems. 1. Use Zorn’s lemma to prove that any submodule of a semisimple module has a complement. 2. Let R be an integral domain and let S be the ring of upper triangular 3 × 3 matrices with entries in R. Find n ≥ 1 and a collection of orthogonal, primitive idempotents e1, . . . , en ∈ S such that e1 + · · · + en = 1S . 3. Let f : V → V be an endomorphism of an n dimensional real vector space such that the endomorphism idC⊗f of the complex vector space C⊗R V is diagonalizable. Prove that there exists a basis for V with respect to which the matrix of the linear transformation f is a block diagonal matrix, with diagonal blocks either being 1 × 1 matrices of the form [λ] for λ ∈ R or being 2× 2 matrices of the form [ r cos θ −r sin θ r sin θ r cos θ ] for r > 0 and 0 < θ < π. When are two such block diagonal matrices similar? 4. Prove that the multiplicative group Cp∞ consisting of all pnth roots of 1 in C× for all n ≥ 0 is an injective abelian group.