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Q.1) Classify the following statements as a true or false statements: a. An ideal n2 is a large ideal in Z. b. Any direct summand of a module M is large in M. c. A module M is called simple if 0 and M are the only submodules of M. d. A monomorphism f: A B is called split iff Im() is a direct summand in M e. The Z-module has two composition series. Q.2) Prove the following: (10 Marks) a) If a: MN is a homomorphism of modules and K is a submodule of M, then a-(a(K)) = K+ Ker(a). b) If M is a simple module and ẞ: MN is a nonzero homomorphism, then N is simple if and only if is an epimorphism. Q.3) Show: (a) Let the diagram be commutative, 1.e. A Ba. Then A- (1) Im(a)+ Ker(B) (Im(A)). (2) Im(a) n Ker(B) = a(Ker(2)). (b) Every factor group of a divisible group is divisible. Q.4) Give an example and explain your claim in each case: a) A monomorphism not split. b) Monomorphism, which is split. c) Semisimple module. λ M d) A large submodule A of a module M and a homomorphism (p: MN, but (A) is not large in N. e) A short split exact sequence of modules. Q.5) Prove the following: a) Every finitely generated submodule of Qz is small in Qz b) Qz does not have a maximal submodules.

Q.1) Classify the following statements as a true or false statements: a. An ideal n2 is a large ideal in Z. b. Any direct summand of a module M is large in M. c. A module M is called simple if 0 and M are the only submodules of M. d. A monomorphism f: A B is called split iff Im() is a direct summand in M e. The Z-module has two composition series. Q.2) Prove the following: (10 Marks) a) If a: MN is a homomorphism of modules and K is a submodule of M, then a-(a(K)) = K+ Ker(a). b) If M is a simple module and ẞ: MN is a nonzero homomorphism, then N is simple if and only if is an epimorphism. Q.3) Show: (a) Let the diagram be commutative, 1.e. A Ba. Then A- (1) Im(a)+ Ker(B) (Im(A)). (2) Im(a) n Ker(B) = a(Ker(2)). (b) Every factor group of a divisible group is divisible. Q.4) Give an example and explain your claim in each case: a) A monomorphism not split. b) Monomorphism, which is split. c) Semisimple module. λ M d) A large submodule A of a module M and a homomorphism (p: MN, but (A) is not large in N. e) A short split exact sequence of modules. Q.5) Prove the following: a) Every finitely generated submodule of Qz is small in Qz b) Qz does not have a maximal submodules.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter6: More On Rings
Section6.4: Maximal Ideals (optional)
Problem 3E
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Q.1) Classify the following statements as a true or false statements: a. An ideal n2 is a large ideal in Z. b. Any direct summand of a module M is large in M. c. A module M is called simple if 0 and M are the only submodules of M. d. A monomorphism f: A B is called split iff Im() is a direct summand in M e. The Z-module has two composition series. Q.2) Prove the following: (10 Marks) a) If a: MN is a homomorphism of modules and K is a submodule of M, then a-(a(K)) = K+ Ker(a). b) If M is a simple module and ẞ: MN is a nonzero homomorphism, then N is simple if and only if is an epimorphism. Q.3) Show: (a) Let the diagram be commutative, 1.e. A Ba. Then A- (1) Im(a)+ Ker(B) (Im(A)). (2) Im(a) n Ker(B) = a(Ker(2)). (b) Every factor group of a divisible group is divisible. Q.4) Give an example and explain your claim in each case: a) A monomorphism not split. b) Monomorphism, which is split. c) Semisimple module. λ M d) A large submodule A of a module M and a homomorphism (p: MN, but (A) is not large in N. e) A short split exact sequence of modules. Q.5) Prove the following: a) Every finitely generated submodule of Qz is small in Qz b) Qz does not have a maximal submodules.
Transcribed Image Text:Q.1) Classify the following statements as a true or false statements: a. An ideal n2 is a large ideal in Z. b. Any direct summand of a module M is large in M. c. A module M is called simple if 0 and M are the only submodules of M. d. A monomorphism f: A B is called split iff Im() is a direct summand in M e. The Z-module has two composition series. Q.2) Prove the following: (10 Marks) a) If a: MN is a homomorphism of modules and K is a submodule of M, then a-(a(K)) = K+ Ker(a). b) If M is a simple module and ẞ: MN is a nonzero homomorphism, then N is simple if and only if is an epimorphism. Q.3) Show: (a) Let the diagram be commutative, 1.e. A Ba. Then A- (1) Im(a)+ Ker(B) (Im(A)). (2) Im(a) n Ker(B) = a(Ker(2)). (b) Every factor group of a divisible group is divisible. Q.4) Give an example and explain your claim in each case: a) A monomorphism not split. b) Monomorphism, which is split. c) Semisimple module. λ M d) A large submodule A of a module M and a homomorphism (p: MN, but (A) is not large in N. e) A short split exact sequence of modules. Q.5) Prove the following: a) Every finitely generated submodule of Qz is small in Qz b) Qz does not have a maximal submodules.
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