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.

  • $\begingroup$ In your second question, presumably you are asking for $K$ to be unique up to isomorphism? As a submodule of $M$, $K$ is almost never unique, even in your examples. $\endgroup$ Feb 29 '16 at 3:43

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 $M/N$).

  • $\begingroup$ Thanks, I think the answer for the first question is adequate. Respect to the second question, when I mentioned "more than one direct complement" it wasn't up to isomorphism, but as sets. for example in $F_{3}^2$ the subspace $V=\{00,12,21\}$ have tree direct complements which are $U_{1}=\{00,11,22\}$, $U_{2}=\{00,10,20\}$, and $U_{3}=\{00,01,02\}$ , so I was looking for an example of a category of $R$-modules, apart from k-vector spaces, in which all $R$-module are injective (i.e. ,$R$ semisimple, as you mentioned) but with possibly more than one complement, as sets. $\endgroup$
    – EGC
    Feb 29 '16 at 5:46
  • $\begingroup$ Complements are almost never unique as sets; this happens over literally every nonzero ring. For instance, taking $M=R^2$ and $N=R\times 0$, both $0\times R$ and $\{(r,r):r\in R\}$ are complements of $N$ in $M$. $\endgroup$ Feb 29 '16 at 5:48
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    $\begingroup$ No, they aren't. For instance, consider the case that $G$ acts trivially on $W$, so submodules of $W$ are just arbitrary vector subspaces. $\endgroup$ Feb 29 '16 at 6:11
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    $\begingroup$ The same thing can happen whenever $W$ contains multiple copies of the same irreducible representation. The example I gave before with $M=R^2$ also works for $M=M_0^2$ for any nonzero module $M_0$. $\endgroup$ Feb 29 '16 at 6:15

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