I am trying to answer Q7. from page 11 of http://www-users.math.umn.edu/~webb/RepBook/RepBookLatex.pdf
The question states:
"Let $G= \langle x, y \ \mid \ x^2=y^2=1 \rangle$ be the Klein four-group, $R=\mathbb{F}_2$ and consider the two representations $\rho_1$ and $\rho_2$ specified on the generators of $G$ by
$$ \rho_1(x) = \begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}, \quad \rho_1(y) = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}, $$ and $$ \rho_2(x) = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix}, \quad \rho_2(y) = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}. $$ Calculate the socles of these two representations. Show that neither representation is semisimple."
The text previous gives the following as the definition of the socle "The largest semisimple submodule of a module $U$ ($A$ is a ring with a $1$, $U$ is an $A$-module of finite composition length) is called the socle of $U$, and is denoted $\operatorname{Soc}(U)$." The text also notes "The sum of all simple submodules of a module is the unique largest semisimple submodule of that module: the socle."
I am having a hard time making sense of this definition and applying it. I am not sure what the submodules are in this case or how to tell what the submodules are in general when given explicit representations.
This is one of the first examples I am considering in this topic so any extra detail you could give me would be helpful and appreciated.