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Consider the category of R-modules, where R is a commutative ring with unity, and let 0 → ABC → 0 be a short exact sequence of R-modules. a) Prove that the sequence is exact if and only if Im(ƒ) = ker(g), ƒ is injective, and g is surjective. Provide a detailed explanation of each implication. b) Assume R is a Noetherian ring and A and C are finitely generated R-modules. Prove that B is also finitely generated. Use the properties of Noetherian rings in your proof. c) Define the notion of a projective module and prove that if A is a projective R-module, then the short exact sequence splits. Provide a comprehensive proof of the splitting. d) Explore the Ext functor by computing Ext (C, A) for the given exact sequence. Explain the relationship between the exact sequence and the Ext group, and determine the conditions under which the sequence splits.

Consider the category of R-modules, where R is a commutative ring with unity, and let 0 → ABC → 0 be a short exact sequence of R-modules. a) Prove that the sequence is exact if and only if Im(ƒ) = ker(g), ƒ is injective, and g is surjective. Provide a detailed explanation of each implication. b) Assume R is a Noetherian ring and A and C are finitely generated R-modules. Prove that B is also finitely generated. Use the properties of Noetherian rings in your proof. c) Define the notion of a projective module and prove that if A is a projective R-module, then the short exact sequence splits. Provide a comprehensive proof of the splitting. d) Explore the Ext functor by computing Ext (C, A) for the given exact sequence. Explain the relationship between the exact sequence and the Ext group, and determine the conditions under which the sequence splits.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter6: More On Rings
Section6.3: The Characteristic Of A Ring
Problem 22E: 22. Let be a ring with finite number of elements. Show that the characteristic of divides .
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Consider the category of R-modules, where R is a commutative ring with unity, and let 0 → ABC → 0 be a short exact sequence of R-modules. a) Prove that the sequence is exact if and only if Im(ƒ) = ker(g), ƒ is injective, and g is surjective. Provide a detailed explanation of each implication. b) Assume R is a Noetherian ring and A and C are finitely generated R-modules. Prove that B is also finitely generated. Use the properties of Noetherian rings in your proof. c) Define the notion of a projective module and prove that if A is a projective R-module, then the short exact sequence splits. Provide a comprehensive proof of the splitting. d) Explore the Ext functor by computing Ext (C, A) for the given exact sequence. Explain the relationship between the exact sequence and the Ext group, and determine the conditions under which the sequence splits.
Transcribed Image Text:Consider the category of R-modules, where R is a commutative ring with unity, and let 0 → ABC → 0 be a short exact sequence of R-modules. a) Prove that the sequence is exact if and only if Im(ƒ) = ker(g), ƒ is injective, and g is surjective. Provide a detailed explanation of each implication. b) Assume R is a Noetherian ring and A and C are finitely generated R-modules. Prove that B is also finitely generated. Use the properties of Noetherian rings in your proof. c) Define the notion of a projective module and prove that if A is a projective R-module, then the short exact sequence splits. Provide a comprehensive proof of the splitting. d) Explore the Ext functor by computing Ext (C, A) for the given exact sequence. Explain the relationship between the exact sequence and the Ext group, and determine the conditions under which the sequence splits.
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