A question about invariant factors

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SUMMARY

The discussion centers on the theorem stating that if R is a Principal Ideal Domain (PID), every finitely generated torsion R-module M can be expressed as a direct sum of cyclic modules. Specifically, M can be represented as M = R/(c_1) ⊕ R/(c_2) ⊕ ... ⊕ R/(c_t), where t ≥ 1 and c_1 | c_2 | ... | c_t. The participants explore methods for determining invariant factors without prior knowledge of elementary divisors, emphasizing the importance of having a presentation of the finitely generated module as a quotient of free modules, which is represented by a matrix that needs to be diagonalized.

PREREQUISITES
  • Understanding of Principal Ideal Domains (PID)
  • Familiarity with finitely generated modules
  • Knowledge of elementary divisors and invariant factors
  • Matrix diagonalization techniques
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  • Study the process of diagonalizing matrices in the context of module theory
  • Research the relationship between elementary divisors and invariant factors in detail
  • Explore examples of finitely generated torsion modules in various PIDs
  • Learn about presentations of modules and their implications in algebraic structures
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Mathematicians, algebraists, and students studying module theory, particularly those focusing on invariant factors and their applications in algebraic structures.

Artusartos
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A Theorem in our textbook says...

If R is a PID, then every finitely generated torision R-module M is a direct sum of cyclic modules

M= R/(c_1) \bigoplus R/(c_2) \bigoplus ... \bigoplus R/(c_t)

where t \geq 1 and c_1 | c_2 | ... | c_t.

There is an example from our textbook that I attached...they find the invariant factors from the elementary divisors. But what if we had to find the invariant factors without being given the elementary divisors. How would we do that?

Thanks in advance
 

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well you have to be given something. i have some examples in my book on my webpage.

to be "given" a f.g. module usually means to be given a "presentation" as a quotient of two free modules.

such a quotient is specified by a matrix. then you diagonalize that presentation matrix.

see the discussion here:

http://www.math.uga.edu/%7Eroy/845-1.pdf
 
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