In page 40 of Auslander‘s representation theory of artin algebra, the proposition 4.1,
For $A$ in mod$\Lambda$ where $\Lambda$ is an artin algebra we have the following.
a) $A=0$ iff $socA=0$
b) A is an essential extension of $socA$.
c) $A\to I$ is n injective envelope of A iff the induced morphism $socA\to I$ is an injective envelope.
d) An injective module I in mod$\Lambda$ is indecomposable iff $socA$ is simple
In the proof of this proposition it is stated that (b),(c) and (d) are easily verified consequences of (a).
But I am confused in using (a) to prove them