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In Exercise 2 of Exercises 1-5 of John Dauns' book "Modules and Rings" we are given
R = \begin{pmatrix} \mathbb{Z}_4 & \mathbb{Z}_4 \\ \mathbb{Z}_4 & \mathbb{Z}_4 \end{pmatrix} and 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:
----------------------------------------------------------------------------------------
Typical elements of R are R = \begin{pmatrix} r_1 & r_2 \\r_3 & r_4 \end{pmatrix}
where 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 x, y \in \overline{2} Z_4
Now the elements of \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 N_1 = \{ (x, 0) | \in \overline{2} |mathbb{Z}_4 - that is x \in \{ 0, 2 \}
Let r \in R and then test the action of R on M i.e. N_1 \times R \rightarrow N_1 - that is test if n_1r |in N_1
Now (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 r_1x and r_2x and hence check whether (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 r_1 = \overline{3} and x = \overline{2} then (roughly speaking!) r_1 x = \overline{3} \overline{2} = \overline{6} = \overline{2}
In the above I am assuming that in \overline{2} \mathbb{Z}_4 that that \overline{0}, \overline{4}, \overline{8}, ... = \overline{0}
and that
that \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
R = \begin{pmatrix} \mathbb{Z}_4 & \mathbb{Z}_4 \\ \mathbb{Z}_4 & \mathbb{Z}_4 \end{pmatrix} and 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:
----------------------------------------------------------------------------------------
Typical elements of R are R = \begin{pmatrix} r_1 & r_2 \\r_3 & r_4 \end{pmatrix}
where 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 x, y \in \overline{2} Z_4
Now the elements of \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 N_1 = \{ (x, 0) | \in \overline{2} |mathbb{Z}_4 - that is x \in \{ 0, 2 \}
Let r \in R and then test the action of R on M i.e. N_1 \times R \rightarrow N_1 - that is test if n_1r |in N_1
Now (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 r_1x and r_2x and hence check whether (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 r_1 = \overline{3} and x = \overline{2} then (roughly speaking!) r_1 x = \overline{3} \overline{2} = \overline{6} = \overline{2}
In the above I am assuming that in \overline{2} \mathbb{Z}_4 that that \overline{0}, \overline{4}, \overline{8}, ... = \overline{0}
and that
that \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