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

and

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

where

Typical elements of M are (x,y) where

Now the elements of

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

Consider a set of the form - that is

Let and then test the action of R on M i.e. - that is test if

Now

But now a problem I hope someone can help with!

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

An example of my thinking here

If and then (roughly speaking!)

In the above I am assuming that in that that

and that

that

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