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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

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    $\begingroup$ Which one do you have trouble with? For example, to prove b), if not, there is a non-zero submodule $M$ of $A$ with $M\cap \mathrm{soc}\, A=0$. But this says $\mathrm{soc}\, M=M\cap\mathrm{soc}\, A=0$ and thus from a), $M=0$. $\endgroup$ – Mohan Nov 9 at 3:34
  • $\begingroup$ @Mohan thanks for your patient answer. Maybe I didn’t understand the socle of a module very well......I’ll rethink about b c and d $\endgroup$ – subHangLou Nov 9 at 4:12

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