Corollary to Correspondence Theorem for Modules

In summary, Peter is seeking help understanding how Corollary 6.25 follows from the Correspondence Theorem for Modules, which can be found in Section 6.1 of Joseph J. Rotman's book Advanced Modern Algebra. The Correspondence Theorem states that submodules of a module $M$ correspond to its intermediate submodules, and when $R$ is considered as a left $R$-module, the submodules of $R$ are equivalent to the left ideals of $R$. Therefore, Corollary 6.25 can be proved by applying the Correspondence Theorem to $R$ itself as a left $R$-module. This will result in the proof of the Corollary.
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I am reading Joseph J. Rotman's book: Advanced Modern Algebra and I am currently focused on Section 6.1 Modules ...

I need some help with the proof of Corollary 6.25 ... Corollary to Theorem 6.22 (Correspondence Theorem) ... ...

Corollary 6.25 and its proof read as follows:View attachment 4925Can someone explain to me exactly how Corollary 6.25 follows from the Correspondence Theorem for Modules ...?

Hope that someone can help ...

Peter=============================================

*** EDIT ***

The above post refers to the Correspondence Theorem for Modules (Theorem 6.22 in Rotman's Advanced Modern Algebra) ... so I am proving the text of the Theorem from Rotman's Advanced Modern Algebra as follows:View attachment 4926
 
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Hint: when $R$ itself is considered as a (left) $R$-module, the submodules of $R$ are precisely the (left) ideals of $R$.
 

FAQ: Corollary to Correspondence Theorem for Modules

What is the Corollary to Correspondence Theorem for Modules?

The Corollary to Correspondence Theorem for Modules is a mathematical statement that describes the relationship between submodules of a module and its quotient modules.

How does the Corollary to Correspondence Theorem for Modules work?

The theorem states that for any submodule N of a module M, there is a one-to-one correspondence between submodules of the quotient module M/N and submodules of M containing N. This means that for every submodule of M/N, there is a corresponding submodule of M containing N, and vice versa.

What is the importance of the Corollary to Correspondence Theorem for Modules?

This theorem is important because it allows us to study quotient modules by looking at their corresponding submodules in the original module. This makes it easier to understand and analyze the structure of quotient modules.

Can the Corollary to Correspondence Theorem for Modules be applied to other mathematical concepts?

Yes, the theorem can also be applied to other algebraic structures such as groups and rings, as long as they have a similar notion of submodules or subgroups. The concept of correspondence between substructures is widely used in mathematics.

Are there any real-life applications of the Corollary to Correspondence Theorem for Modules?

The theorem has many applications in various fields such as physics, computer science, and engineering. It is particularly useful in coding theory, where it is used to study error-correcting codes. It also has applications in signal processing and control theory.

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