Injective Modules Motivation & Intuition
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?
Solution 1:
I've always thought of injective modules this way (beyond simply a dual definition to projective modules):
- Injective modules are a summand of any module containing them
- Every module embeds in an injective module, which is a kind of "completion"
Now, I have not worked with cofree modules, but you may want to check out this entry on cofree modules. On one hand, I believed that such a dual description exists, on the other hand, I don't have personal experience with it, and I don't find wolfram mathworld to be very reliable. So please take it with a grain of salt.
It's also worth noting that the second bullet above does not dualize: the dual would usually be considered to be "every module has a projective cover" but it is not true. However, a famous result is that every module has a flat cover. Rings for which every right module has a projective cover are known as right perfect rings.