A module $M$ over a commutative ring $R$ is called a 'injective module' if it satisfies certain universal property explaned here.
Question: Is there any intuition how to think concretely about injective modules? Do them naturally arise as an attempt go generalize a special class of modules? I'm asking this because I try to find an analogy to the dual concept of projective modules.
Although these are formally defined by a similar (but dual) UP these have a more accessible interpretation: These arise as a natural generalization of free modules and form literally finer building blocks of free modules since there is a fact that a module is projective iff it is a direct summand of a free module.
Does these exist a similar interpretation for injective modules? Which class of modules do these naturally generalize and do they arise also as 'building blocks' of something?