R Module M is Cyclic: Isomorphic to R/(p)?

Thread starter pivoxa15
Start date May 10, 2007
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Cyclic module

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SUMMARY

If an R module M is cyclic and the annihilator of m is (p), where p is a prime, then M is isomorphic to R/(p). This conclusion is drawn from the properties of cyclic modules and their relationship with annihilators. Additionally, a linear transformation T defines a cyclic F[λ] module if and only if the characteristic polynomial of T equals its minimum polynomial. This establishes a clear criterion for cyclicity in the context of linear transformations.

PREREQUISITES

Understanding of R modules and their properties
Knowledge of annihilators in module theory
F[λ] modules and linear transformations
Familiarity with characteristic and minimum polynomials

NEXT STEPS

Study the structure of cyclic R modules
Explore the concept of annihilators in module theory
Learn about the relationship between characteristic and minimum polynomials
Investigate examples of linear transformations and their corresponding F[λ] modules

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Mathematicians, graduate students in algebra, and anyone studying module theory and linear algebra concepts.

May 10, 2007

pivoxa15

Homework Statement

If an R module M is cyclic so M=Rm with annihilator(m)=(p), p prime

then can we infer that M is isomorphic to R/(p) without any more infomation?

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May 8, 2008

mobe

Is it true that the F[\lambda] module determined by a linear transformation T is
cyclic iff the characteristic polynomial of T equals the minimum polynomial
of T?