Corollary to Correspondence Theorem for Modules

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SUMMARY

The discussion centers on the proof of Corollary 6.25, which is derived from Theorem 6.22, the Correspondence Theorem for Modules, as presented in Joseph J. Rotman's book, Advanced Modern Algebra. The key insight is that when the ring R is viewed as a left R-module, its submodules correspond directly to the left ideals of R. This establishes a clear link between the concepts of modules and ideals, facilitating a deeper understanding of module theory.

PREREQUISITES
  • Understanding of module theory, specifically the Correspondence Theorem for Modules.
  • Familiarity with left R-modules and their properties.
  • Knowledge of ideals in ring theory.
  • Basic proficiency in algebraic structures as outlined in Rotman's Advanced Modern Algebra.
NEXT STEPS
  • Study the proof of Theorem 6.22 in Rotman's Advanced Modern Algebra.
  • Explore the relationship between submodules and ideals in more depth.
  • Investigate additional corollaries related to the Correspondence Theorem for Modules.
  • Practice problems involving the application of the Correspondence Theorem in various algebraic contexts.
USEFUL FOR

Mathematics students, algebra researchers, and educators seeking to deepen their understanding of module theory and its applications in advanced algebra.

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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 $R$.
 

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