In the book "Modules and Rings" by John Dauns he adopts the following notation in Exercises 1-5 (see attached)

"For additive groups A, B, C, D and $\displaystyle a \in A, b \in B, c \in C, d \in D $ we write

$\displaystyle [a,b; c,d] = \begin {vmatrix} a & b \\ c & d \end {vmatrix} $ ; $\displaystyle [A, B, C, D] = \begin {vmatrix} A & B \\ C & D \end {vmatrix} $ "

Exercise 1 (i), (ii) and (iii)

The matrix ring R acts on the right R-module M whose elements are row vectors:

Find (i) all submodules of M (ii) all right ideals of R, indicating which of these are ideals

$\displaystyle R = [ \mathbb{Z}_2, \mathbb{Z}_2; 0, \mathbb{Z}_2, ] , M = \mathbb{Z}_2 \oplus \mathbb{Z}_2, $

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Would be most grateful for help

Peter