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- Jun 22, 2012

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[TEX] R = \begin{pmatrix} \mathbb{Z}_4 & \mathbb{Z}_4 \\ \mathbb{Z}_4 & \mathbb{Z}_4 \end{pmatrix} [/TEX] and [TEX]M = \overline{2} \mathbb{Z}_4 \times \overline{2} \mathbb{Z}_4 [/TEX]

where the matrix ring R acts on a right R-module whose elements are row vectors.

**Find all submodules of M**

Helped by Evgeny's post (See Math Help Boards) I commenced this problem as follows:

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Typical elements of R are [TEX] R = \begin{pmatrix} r_1 & r_2 \\r_3 & r_4 \end{pmatrix} [/TEX]

where [TEX] r_1, r_2, r_3, r_4 \in \mathbb{Z}_4 = \{ \overline{0}, \overline{1}, \overline{2}, \overline{3} \} [/TEX]

Typical elements of M are (x,y) where [TEX] x, y \in \overline{2} Z_4 [/TEX]

Now the elements of [TEX] \overline{2} Z_4 = \overline{2} \times \{ \overline{0}, \overline{1}, \overline{2}, \overline{3} \} = \{ \overline{0}, \overline{2}, \overline{4}, \overline{6} \} = \{ \overline{0}, \overline{2}, \overline{0}, \overline{2} \} = \{ \overline{0}, \overline{2} \} [/TEX]

Now to find submodules! (Approach is by trial and error - but surely there is a better way!)

Consider a set of the form [TEX] N_1 = \{ (x, 0) | \in \overline{2} |mathbb{Z}_4 [/TEX] - that is [TEX] x \in \{ 0, 2 \} [/TEX]

Let [TEX] r \in R [/TEX] and then test the action of R on M i.e. [TEX] N_1 \times R \rightarrow N_1 [/TEX] - that is test if [TEX] n_1r |in N_1 [/TEX]

Now [TEX] (x, 0) \begin{pmatrix} r_1 & r_2 \\r_3 & r_4 \end{pmatrix} = (r_1x, r_2x ) [/TEX]

**But now a problem I hope someone can help with!**How do we (rigorously) evaluate [TEX] r_1x [/TEX] and [TEX] r_2x [/TEX] and hence check whether [TEX] (r_1x, r_2x) [/TEX] is of the form (x, 0) [certainly does not look like it but formally and rigorously ...?]

An example of my thinking here

If [TEX] r_1 = \overline{3} [/TEX] and [TEX] x = \overline{2} [/TEX] then (roughly speaking!) [TEX] r_1 x = \overline{3} \overline{2} = \overline{6} = \overline{2}[/TEX]

In the above I am assuming that in [TEX] \overline{2} \mathbb{Z}_4 [/TEX] that that [TEX] \overline{0}, \overline{4}, \overline{8}, ... = \overline{0} [/TEX]

and that

that [TEX] \overline{2}, \overline{6}, \overline{10}, ... = \overline{2} [/TEX]

but I am not sure what I am doing here!

**Can someone please clarify this situation?**

Further, can someone please comment on my overall approach to the Exercise - I am not at all sure regarding how to check for submodules and certainly lack a systematic approach ...

Be grateful for some help ...Further, can someone please comment on my overall approach to the Exercise - I am not at all sure regarding how to check for submodules and certainly lack a systematic approach ...

Be grateful for some help ...

Peter