Let $R$ a ring (non conmutative in general) with unit. We say that $R$ is a $QI$-Ring (quasi-injective ring) iff for every $R$-mod $Q$, QI (quasi-injective module) is also injective.
I need prove the following equivalence:
$R$ is $QI$-ring.
The direct sum of any two $QI$ $R$-modules are a $QI$.
I know that every injective module is $QI$, so (1)$\Rightarrow$(2) is obvious. But (2)$\Rightarrow$(1) it's hard to see for me. I know that any $QI$ module that contains a copy of regular module $R_R$ is injective and in general the sum of QI modules is not in general QI.
I've been tried to find more info about $QI$ rings but I don't find any specific, also I know that every $QI$ rings is also an hereditary noetherian V-right but i'm searching for a direct proof using only that $R$ is $QI$ ring and properties of $QI$ modules.
Any hint or source with info of $QI$ rings? (I've read "Algebra" of Carl Faith but the only usefull references are that "If $A\oplus B$ is a $QI$ module if and only if $A$ and $B$ are injective in $R$-mod".)
If $M$ is a quasi injective module and your cited proposition is true, and you assume 2), then $M\oplus M$ is quasi injective by hypothesis, and then $M$ is injective by the proposition. That proves 1).