Let (R,+..) be a ring of real numbers and (Max+..) is a ring of 2×2 matrices over R. Let f:R¬such that (a)(). ThenKa) fis an isomorphism.b) fis not one-to-one but onto homomorphism.5377c) fis a homomorphism but not onto and not one-to-one.d) F is one-to-one but not onto homomorphism.Let f(x), g(x) € Z(x) such that f(x)=1+2x+3x² and g(x)=2+x² + 2x³. Thena) deg(f(x)g(x)) < deg(f(x)) + deg(g(x)).b) deg(f(x)g(x)) = 0.BOc) deg(f(x)g(x)) deg(f(x)) + deg(g(x)).d) deg(f(x)g(x))> deg(f(x)) + deg(g(x)).Let V be a finite dimension vector space over a field F and S₁, S are subsets of V such that S, is aper subset of S₂ Thena) If S₂ be a basis for V then S, is a basis for V.b) If S, be a basis for V then Sy is a basis for V.c) If S, generate V then S, generate V.d) If Sa generate V then S₁ generate V.

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Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter6: More On Rings

Section6.2: Ring Homomorphisms

Problem 18E

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Let (R,+..) be a ring of real numbers and (Max+..) is a ring of 2×2 matrices over R. Let f:R¬ such that (a)(). Then Ka) fis an isomorphism. 557 b) fis not one-to-one but onto homomorphism. c) fis a homomorphism but not onto and not one-to-one. d) Fis one-to-one but not onto homomorphism.