I'm working through Isaacs' Algebra- A Graduate Course, and I have hit a wall at the chapter on Modules. In particular I am having trouble with Problem 12.24.
Say $B, C \subseteq A$ are unitary subrings of $A$. Let $A_B$ and $_CA$ denote $A$ viewed as a right $B$-module and $A$ viewed as a left $C$-module, respectively. Also, let $$R =\left\{ \begin{bmatrix} b & 0 \\ a & c \end{bmatrix} \middle\vert a \in A, b \in B, c \in C \right\} \subseteq M_2(A)$$
and let $$I = \left\{ \begin{bmatrix} 0 & 0 \\ a & 0 \end{bmatrix} \middle\vert a \in A \right\} \subseteq R.$$
The first two parts of the question have us proving that $R$ is a ring and $I$ is an ideal of $R$, then it asks us to show that $R/I \cong B \oplus C$. So those facts are available to us.
We are then asked to show that $I$ is noetherian or artinian as a right $R$-module if and only if $A_B$ is noetherian or artinian, respectively.
In order for $I$ to be noetherian or artinian, it must be an abelian $X$-group and we assign a poset $P$ consisting all $X$-subgroups of $I$ ordered by inclusion. Going by the definition of $X$-subgroup, $H \subseteq G$ is an $X$-subgroup if for every $x \in X$ and $h \in H$, we have $h^{x} \in H$.
It seems to me that every matrix $x \in I$ is singular. So how is it possible for $h^x = x^{-1}hx \in H \subseteq X$? What exactly do the $X$-subgroups "look like" ?