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

Thread starter pivoxa15
Start date May 10, 2007
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Cyclic module

In summary, a cyclic module is a module that can be generated by a single element and can be written as a linear combination of the generator. To determine if a module is cyclic, there must exist a single element that can generate the entire module. If a cyclic module is isomorphic to R/(p), it means that there is a bijective homomorphism between the two structures, showing that their structures are identical and their elements can be mapped to each other. To prove this isomorphism, a bijective homomorphism must be explicitly defined and shown to be injective and surjective. The significance of a module being isomorphic to R/(p) lies in understanding the properties and connections between the two structures.

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[[tex]\lambda[/tex]] module determined by a linear transformation T is
cyclic iff the characteristic polynomial of T equals the minimum polynomial
of T?

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

1. What is the definition of a cyclic module?

A cyclic module is a module that can be generated by a single element. This means that every element in the module can be written as a linear combination of the generator.

2. How can I determine if a module M is cyclic?

A module M is cyclic if and only if there exists a single element m in M such that every element in M can be written as a linear combination of m. In other words, if M is generated by a single element, then it is cyclic.

3. What does it mean for a cyclic module to be isomorphic to R/(p)?

If a cyclic module M is isomorphic to R/(p), it means that there exists a bijective homomorphism between M and R/(p). This essentially means that the structure of M is identical to the structure of R/(p), and the elements in M can be mapped to corresponding elements in R/(p).

4. How can I prove that a cyclic module M is isomorphic to R/(p)?

To prove that a cyclic module M is isomorphic to R/(p), you must show that there exists a bijective homomorphism between the two structures. This can be done by explicitly defining a homomorphism and showing that it is both injective and surjective.

5. What is the significance of a module being isomorphic to R/(p)?

The fact that a cyclic module M is isomorphic to R/(p) tells us that the structure of M is essentially the same as the structure of the quotient ring R/(p). This can be useful in understanding the properties of M and making connections between the two structures.