Let $D$ be a division ring and let $_D V_D$ be a $D$-$D$-bimodule. If we temporarily forget the left module structure, and just look at the right $D$-module structure, we have $V_D= \bigoplus_{i\in I}e_iD$ for some basis $\{e_i\}_{i\in I}$.

It is a well-known fact that $E:={\rm End}(V_D)\cong {\rm CFM}_I(D)$ where ${\rm CFM}_I(D)$ is the ring of $I\times I$ column finite matrices. These are the matrices where each column has only finitely many nonzero entries. If we think of the elements of $V_D$ as columns of size $I\times 1$ with only finitely many nonzero entries, then ${\rm CFM}_I(D)$ acts on the left of $V_D$ simply by matrix multiplication. (Of course, when $|I|=n$ is finite, then ${\rm CFM}_I(D)=\mathbb{M}_n(D)$ is just the usual ring of $n\times n$ matrices.)

So we have a natural bimodule structure on $V$, namely $_{E}V_D$. Our original bimodule structure $_DV_D$ gives rise to a homomorphism $\varphi:D\to E\cong {\rm CFM}_I(D)$. Conversely, given any such homomorphism (and a fixed basis for $V_D$ indexed by $I$) we get a $D$-$D$-module structure on $V$.

We could do all of this over again on the other side. From the left $D$-module structure $_DV$, we can fix a basis $\{f_j\}_{j\in J}$ and corresponding decomposition $_DV=\bigoplus_{j\in J}Df_j$. The right $D$-module structure then corresponds to a homomorphism $\psi:D\to {\rm RFM}_J(D)$. (The ring ${\rm RFM}_J(D)$ is the ring of $J\times J$ row finite matrices.)

So given an index set $I$ and a homomorphism $\varphi:D\to {\rm CFM}_I(D)$, there is a corresponding index set $J$ and a homomorphism $\psi:D\to {\rm RFM}_J(D)$. My question is whether there is a canonical way to describe the correspondence $(I,\varphi)\leftrightarrow(J,\psi)$. If not a canonical way, given the information $I$ and $\varphi$, can we at least describe $|J|$ and $\psi$ explicitly from that data, after a choice of basis?

If $_DV_D$ is a $D$-$D$-bimodule, and we have a $D$-basis for $V_D$, do we still need AC to get a $D$-basis for $_DV$?

(The original question appears below. But this shorter question gets at the heart of my question, and makes it clear it has more logical foundations.)

Let $D$ be a division ring and let $_D V_D$ be a $D$-$D$-bimodule. If we temporarily forget the left module structure, and just look at the right $D$-module structure, we have $V_D= \bigoplus_{i\in I}e_iD$ for some basis $\{e_i\}_{i\in I}$.

It is a well-known fact that $E:={\rm End}(V_D)\cong {\rm CFM}_I(D)$ where ${\rm CFM}_I(D)$ is the ring of $I\times I$ column finite matrices. These are the matrices where each column has only finitely many nonzero entries. If we think of the elements of $V_D$ as columns of size $I\times 1$ with only finitely many nonzero entries, then ${\rm CFM}_I(D)$ acts on the left of $V_D$ simply by matrix multiplication. (Of course, when $|I|=n$ is finite, then ${\rm CFM}_I(D)=\mathbb{M}_n(D)$ is just the usual ring of $n\times n$ matrices.)

So we have a natural bimodule structure on $V$, namely $_{E}V_D$. Our original bimodule structure $_DV_D$ gives rise to a homomorphism $\varphi:D\to E\cong {\rm CFM}_I(D)$. Conversely, given any such homomorphism (and a fixed basis for $V_D$ indexed by $I$) we get a $D$-$D$-module structure on $V$.

We could do all of this over again on the other side. From the left $D$-module structure $_DV$, we can fix a basis $\{f_j\}_{j\in J}$ and corresponding decomposition $_DV=\bigoplus_{j\in J}Df_j$. The right $D$-module structure then corresponds to a homomorphism $\psi:D\to {\rm RFM}_J(D)$. (The ring ${\rm RFM}_J(D)$ is the ring of $J\times J$ row finite matrices.)

So given an index set $I$ and a homomorphism $\varphi:D\to {\rm CFM}_I(D)$, there is a corresponding index set $J$ and a homomorphism $\psi:D\to {\rm RFM}_J(D)$. My question is whether there is a canonical way to describe the correspondence $(I,\varphi)\leftrightarrow(J,\psi)$. If not a canonical way, given the information $I$ and $\varphi$, can we at least describe $|J|$ and $\psi$ explicitly from that data, after a choice of basis?