In Exercise 2 of Exercises 1-5 of John Dauns' book "Modules and Rings" we are given

$\displaystyle R = \begin{pmatrix} \mathbb{Z}_4 & \mathbb{Z}_4 \\ \mathbb{Z}_4 & \mathbb{Z}_4 \end{pmatrix} $ and $\displaystyle M = \overline{2} \mathbb{Z}_4 \times \overline{2} \mathbb{Z}_4 $

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 $\displaystyle R = \begin{pmatrix} r_1 & r_2 \\r_3 & r_4 \end{pmatrix} $

where $\displaystyle r_1, r_2, r_3, r_4 \in \mathbb{Z}_4 = \{ \overline{0}, \overline{1}, \overline{2}, \overline{3} \} $

Typical elements of M are (x,y) where $\displaystyle x, y \in \overline{2} Z_4 $

Now the elements of $\displaystyle \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} \} $

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

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

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

Now $\displaystyle (x, 0) \begin{pmatrix} r_1 & r_2 \\r_3 & r_4 \end{pmatrix} = (r_1x, r_2x ) $

But now a problem I hope someone can help with!

How do we (rigorously) evaluate $\displaystyle r_1x $ and $\displaystyle r_2x $ and hence check whether $\displaystyle (r_1x, r_2x) $ is of the form (x, 0) [certainly does not look like it but formally and rigorously ...?]

An example of my thinking here

If $\displaystyle r_1 = \overline{3} $ and $\displaystyle x = \overline{2} $ then (roughly speaking!) $\displaystyle r_1 x = \overline{3} \overline{2} = \overline{6} = \overline{2}$

In the above I am assuming that in $\displaystyle \overline{2} \mathbb{Z}_4 $ that that $\displaystyle \overline{0}, \overline{4}, \overline{8}, ... = \overline{0} $

and that

that $\displaystyle \overline{2}, \overline{6}, \overline{10}, ... = \overline{2} $

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 ...

Peter