Given a bound quiver $(Q, I)$ and a representation $M$ of $Q$, how to get the injective envelope and projective cover of $M$? how to give the corresponding essential monomorphism and superfluous epimorphism? Is there a general or specific method?
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I assume you know the notations $S(i), P(i), I(i)$ for the simple, projective, and injective module (respectively) associated to the vertex $i$. Each of these modules is easily definable so any text on the module theory of path algebras and their quotients will have their definitions.
I will also assume you know how to find the radical and the socle of any given module. Again this is something standard that you'll find in any text on this stuff. You are given a module as a representation of the quiver and you write down a new representation according to a few basic rules.
Now some facts:
And that's all you need to know. So for example if you want the essential surjection from the projective cover of $M$ to $M$ you do the following:
To get injective envelopes it's the exact same set of steps, just dualized so that you're computing injections instead of surjections.