# socle and essential extension of finitely generated mod over an artin algebra

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

• 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), $M=0$. – Mohan Nov 9 at 3:34
• @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 – subHangLou Nov 9 at 4:12