Thread: Artinian Modules - Cohn Exercise 3, Section 2.2, page 65

In summary, the conversation revolves around the book "Introduction to Ring Theory" by P.M. Cohn, specifically Section 2.2 which discusses Artinian and Noetherian rings and modules. The person is seeking help with Exercise 3 on page 65, which involves considering submodules and quotient modules in relation to endomorphisms. Atiyah and Macdonald provide hints for both the Noetherian and Artinian cases.
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I am reading P.M. Cohn's book: Introduction to Ring Theory (Springer Undergraduate Mathematics Series) ... ...

I am currently focused on Section 2.2: Chain Conditions ... which deals with Artinian and Noetherian rings and modules ... ...

I need help to get started on Exercise 3, Section 2.2, page 65 ...

Exercise 3 (Section 2.2, page 65) reads as follows:
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Any help will be much appreciated ...

Peter
 

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The same exercise is in my book (see your other thread). There Atiyah, Macdonald give the following hints:
Let ##φ : M → M## be the endomorphism. To prove the Noetherian case consider the submodules ##ker(φ^n)## and in the Artian case the quotient modules ##coker(φ^n) = M / im(φ^n) , n ∈ℕ.##
 
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Related to Thread: Artinian Modules - Cohn Exercise 3, Section 2.2, page 65

1. What are Artinian modules?

Artinian modules are modules that satisfy the descending chain condition on submodules. This means that for every descending chain of submodules M1 ⊇ M2 ⊇ ..., there exists some positive integer n such that Mn = Mn+1 = ... = 0. In other words, there is a finite number of submodules in the chain before reaching the zero submodule.

2. What is Cohn Exercise 3 in Section 2.2, page 65?

Cohn Exercise 3 is a problem from the book "Basic Algebra: Groups, Rings, and Fields" by P.M. Cohn. It is located in Section 2.2, page 65, and it deals with Artinian modules and their properties.

3. How is Exercise 3 in Section 2.2 related to Artinian modules?

Cohn Exercise 3 in Section 2.2 is directly related to Artinian modules because it asks the reader to prove various properties of Artinian modules, such as the fact that every submodule of an Artinian module is also Artinian.

4. What is the purpose of Exercise 3 in Section 2.2?

The purpose of Exercise 3 in Section 2.2 is to help the reader gain a better understanding of Artinian modules and their properties. By working through the exercises, the reader can practice applying the definitions and theorems related to Artinian modules and strengthen their knowledge in this area.

5. Are Artinian modules important in the field of mathematics?

Yes, Artinian modules are important in the field of mathematics because they are used to study and classify various mathematical objects, such as rings and algebras. They also have applications in algebraic geometry, representation theory, and more. Understanding Artinian modules can lead to a deeper understanding of these areas of mathematics.

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