I have some questions I would like to answer. Both of them are related to injective modules. First, I want to clarify that I am not interested in the following categories as options to answer my questions: k-vectorial spaces and $F[G]$-modules in the semisimple case (when $(|G|,Char(F))=1$).
I am looking for categories of $R$-modules in which every module in injective,
Do you know a category of left $R$-modules with that property?
The following question is related to this equivalence of been injective: Let $Q$ a left $R$-module, $Q$ is injective if for all left $R$-module $M$ such that $Q$ is a submodule of $M$, there exists another submodule $K$ of $M$ such that $M$ is the internal direct sum of $Q$ and $K$, i.e. $M=Q\oplus K$. So, my question is, Are there examples of injective modules $Q$ with more than one direct complement $K$? Moreover, There exist a category of left $R$-modules such that every left $R$-module $Q$ is injective, but not all $R$-module have a unique direct complement?
I know that, if $R$ is a field or the group algebra $F[G]$ when $(|G|,Char(F))=1$, every left $R$-module is injective, and especially when $R$ is a field the second and third question have affirmative answers.
Every $R$-module is injective iff $R$ is a semisimple ring, and by the Artin-Wedderburn theorem a ring is semisimple iff it is isomorphic to a finite product $\prod M_{n_i}(D_i)$ of matrix rings over division rings. For your second question, whenever you have a module $M$ over any ring and a submodule $N\subseteq M$ which has a complement, all the complements $K$ of $N$ are isomorphic, since they are all isomorphic to the quotient module $M/N$ (the quotient map $M=N\oplus K\to M/N$ restricts to an isomorphism from $K$ to $M/N$).