Z[i]-module Question: Finding Torsion and Integer r for Module M

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Z-module Question

Let [tex]M[/tex] be the [tex]\mathbb{Z}[/tex]-module generated by the elements [tex]v_1[/tex], [tex]v_2[/tex] such that [tex](1+i)v_1+(2-i)v_2=0[/tex] and [tex]3v_1+5iv_2=0[/tex]. Find an integer [tex]r \geq 0[/tex] and a torsion [tex]\mathbb{Z}[/tex]-module [tex]T[/tex] such that [tex]M \cong \mathbb{Z}^r \times T[/tex].
 
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Can you find the torsion submodule? After that, Zr should be isomorphic to M/T.
 

FAQ: Z[i]-module Question: Finding Torsion and Integer r for Module M

What is a Z[i]-module?

A Z[i]-module is a module over the ring of Gaussian integers, which is the set of numbers of the form a + bi, where a and b are integers and i is the imaginary unit. This ring is similar to the ring of integers, but includes imaginary numbers as well.

What is torsion in a Z[i]-module?

Torsion in a Z[i]-module refers to elements that become zero after multiplication by some integer r. In other words, for an element m in the module M, there exists an integer r such that rm = 0. Torsion elements play an important role in understanding the structure of Z[i]-modules.

How do you find torsion in a Z[i]-module?

To find torsion elements in a Z[i]-module, you can use the fact that a Z[i]-module is also a vector space over the field of complex numbers. This means that you can use linear algebra techniques, such as finding the kernel of a linear transformation, to identify torsion elements.

What is the significance of integer r in a Z[i]-module?

The integer r in a Z[i]-module is significant because it determines the order of the torsion element. In other words, if rm = 0 for some element m in the module, then r is the smallest positive integer such that rm = 0. This information is useful in understanding the structure and properties of the module.

How can you use the concept of torsion to classify Z[i]-modules?

Torsion elements play a crucial role in classifying Z[i]-modules. For example, a Z[i]-module is torsion-free if it does not contain any nonzero torsion elements. This means that the module behaves like a vector space over the field of complex numbers. On the other hand, a Z[i]-module with nonzero torsion elements has a more complicated structure and can be further classified based on the orders of these torsion elements.

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