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91 ***** 14/12/2021 Q1) a) Let a: AB be a homomorphism, then a is a surjection a is an epimorphism. b) If A-a B *** Y C- δ D Se is commutative, i.e. ẞa=8y and if y is epimorphism, and ẞ is monomorphism, then we have (a) Im(a) = B1(Ims), (b) Ker(8) = y(Kera). Q2) a) Let M be the ZZ-module Z 40Z 1) Find two composition series of M. (1) 2) Explain the isomorphism between the series in (4). 3) Determine the length of M. b) Prove: Qz has no minimal and no maximal submodules. Q3) a) Define finitely generated and finitely cogenerated module. Then give positive and negative examples. b) Prove: If the module MR if finitely generated then every proper submodule of M is contained in a maximal submodule of M. 5 din (v) = 20-

91 ***** 14/12/2021 Q1) a) Let a: AB be a homomorphism, then a is a surjection a is an epimorphism. b) If A-a B *** Y C- δ D Se is commutative, i.e. ẞa=8y and if y is epimorphism, and ẞ is monomorphism, then we have (a) Im(a) = B1(Ims), (b) Ker(8) = y(Kera). Q2) a) Let M be the ZZ-module Z 40Z 1) Find two composition series of M. (1) 2) Explain the isomorphism between the series in (4). 3) Determine the length of M. b) Prove: Qz has no minimal and no maximal submodules. Q3) a) Define finitely generated and finitely cogenerated module. Then give positive and negative examples. b) Prove: If the module MR if finitely generated then every proper submodule of M is contained in a maximal submodule of M. 5 din (v) = 20-

Elements Of Modern Algebra
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 11E: 11. Show that defined by is not a homomorphism.
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91 ***** 14/12/2021 Q1) a) Let a: AB be a homomorphism, then a is a surjection a is an epimorphism. b) If A-a B *** Y C- δ D Se is commutative, i.e. ẞa=8y and if y is epimorphism, and ẞ is monomorphism, then we have (a) Im(a) = B1(Ims), (b) Ker(8) = y(Kera). Q2) a) Let M be the ZZ-module Z 40Z 1) Find two composition series of M. (1) 2) Explain the isomorphism between the series in (4). 3) Determine the length of M. b) Prove: Qz has no minimal and no maximal submodules. Q3) a) Define finitely generated and finitely cogenerated module. Then give positive and negative examples. b) Prove: If the module MR if finitely generated then every proper submodule of M is contained in a maximal submodule of M. 5 din (v) = 20-
Transcribed Image Text:91 ***** 14/12/2021 Q1) a) Let a: AB be a homomorphism, then a is a surjection a is an epimorphism. b) If A-a B *** Y C- δ D Se is commutative, i.e. ẞa=8y and if y is epimorphism, and ẞ is monomorphism, then we have (a) Im(a) = B1(Ims), (b) Ker(8) = y(Kera). Q2) a) Let M be the ZZ-module Z 40Z 1) Find two composition series of M. (1) 2) Explain the isomorphism between the series in (4). 3) Determine the length of M. b) Prove: Qz has no minimal and no maximal submodules. Q3) a) Define finitely generated and finitely cogenerated module. Then give positive and negative examples. b) Prove: If the module MR if finitely generated then every proper submodule of M is contained in a maximal submodule of M. 5 din (v) = 20-
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