Isomorphism classes of simple right modules over a matrix ring.

Question

Let $A\subset M_2(k)$ for a field $k$, be a matrix subring consisting of matrices of the form $$A=\left(\begin{matrix}*&*\\0&* \end{matrix} \right). $$ Then, consider the maximal right ideals $$m_1=\left( \begin{matrix}1&1\\0&0 \end{matrix} \right) \text{ and } m_2=\left( \begin{matrix}0&1\\0&1 \end{matrix}\right).$$ Then, $A/m_1 \cong k$ with right action defined by $$1 \cdot \left(\begin{matrix}a&b\\0&c \end{matrix} \right)=b. $$ Additionally, $A/m_2 \cong k$ with right action defined by $$1 \cdot \left(\begin{matrix}a&b\\0&c \end{matrix} \right)=a. $$ However, these are not isomorphic modules, despite them both being isomorphic to $k$. Indeed, let $$f:k \to k $$ be non trivial. Then, if the module were isomorphic, $$f\left(1 \cdot \left(\begin{matrix}a&b\\0&c \end{matrix} \right) \right)=f(1)\left( \begin{matrix}a&b\\0&c \end{matrix} \right), $$ $$\Rightarrow f(b)=f(1)\cdot a $$ which clearly can't hold for every $a,b \in k$, and so the modules cannot be isomorphic.

$\textbf{Finally, my question:}$ When we consider the isomorphism classes of simple right modules (both of the right modules presented above are simple), what are we to do about the fact that both of the modules are isomorphic to $k$, but not to each other as modules. This is simply a consequence of how we defined the action on $A$ in their cases, and so my intuition is to say they both fall into the same isomorphism class, but I'm not actually sure whether this is correct. What's the standard convention in this case?

Answer 1

what are we to do about the fact that both of the modules are isomorphic to $$, but not to each other as modules.

Here might be a good time to pause and ask "isomorphic to $k$... as what type of object?" An isomorphism is always in the context of a particular category. This case is a good example: the two modules are isomorphic as sets, but not isomorphic as modules.

and so my intuition is to say they both fall into the same isomorphism class, but I'm not actually sure whether this is correct.

Modules can't be both nonisomorphic and fall into the same isomorphism class, no. The two contradict each other.

Perhaps you are just experiencing a little cognitive dissonance around one set $k$ having disparate module structures?

The two structures are completely different. $\begin{bmatrix}0&k\\ 0&0\end{bmatrix}$ annihilates the first but not the second, and $\begin{bmatrix}k&0\\ 0&0\end{bmatrix}$ annihilates the second and not the first. The two module structures behave completely differently. Of course, isomorphic modules have equal annihilators. The two are not in the same isomorphism class of $R$ modules.

But as sets, isomorphism just means a bijection, so the two are in the same isomorphism class of sets.