While trying to find structures that would solve a problem I have to make a replayable detective type of book, I have run into a set of properties that would solve my problems, if only I could find examples of objects that have such properties, and this led me to writing this question.
Let $X \in \mathcal{M}_{m \times n} (M)$, and $B$ be an $n \times m$ matrix over some group $G$.
If I believe correctly, the product $XB$ would require $G$ to be an $M$-module to be well defined, and the resulting matrix $C$ would be in $\mathcal{M}_{m \times m} (G)$. Each $c_{i,j}$ would then be of the form $$c_{i,j} = \sum_{k=1}^m x_{i,k}b_{k,j}.$$
What are the necessary conditions for the product $XB$ to ''uniquely define $C$ on the left''? That is, if $X_1B = X_2B$, then $X_1 = X_2$.
Is there a way to also have $c_{i,j}$ to be uniquely determined by the sum that defines it? By that I mean, is there a way to have $\sum_{p=1}^m a_{i,p}b_{p,j} = \sum_{q=1}^m a_{i,q}b_{q,j}$ if, and only if, $a_{i,p} = a_{i,q}$ and $b_{p,j} = b_{q,j}$ for all $q \in \{1,\ldots,m\}$.
I imagine this uniqueness would pose conditions onto both $X$ and $G$, although I don't even know if it possible to have this while still having the first condition.
If the question is too vague to be understood, I would love to try to reformulate it giving additional context.