Is there a name for such class of modules $M$ such that $M\rightarrow N\rightarrow 0$ splits for every $N$?



Projectively splitting module
These are the [semisimple modules](https://en.wikipedia.org/wiki/Semisimple_module). ("Semisimple" is usually defined by the condition that every injection into M splits, but that's clearly equivalent to your condition.)


