Please ignore, I have managed to solve the question (Length of finitely generated module over $0$-dimensional Gorenstein local ring)

Please see attached images: Can someone slowly walk me through the details of the proof of Lemma 4.1 using basic commutative algebraic arguments and definitions?

1) Why $R[m]?k$? (where $k$ is the residue field of $R$)
2) I know length is additive on direct sums, but how is the RHS of the inequality additive as well, since it is a product?
3) Why does the inequality coincide when $R$ is the module over itself?
4) How does $R$ being Gorenstein of dimension $0$ imply that $R$ is the only indecomposable injective module, and why is its injective hull free as a result?
5) Why $M \subseteq \mathfrak{m}F$?
6) Why does $M=0$ imply the desired result?

  • I might be able to give it a look next week, if you extend the deadline. This week I'm on a work trip unfortunately.

  • Yeah sure, I'm gonna extend by 1 more week, thanks!

  • Cool, I'll give it a look next Monday, I hope I can get sense of it!

  • Hey sorry, this week is being bad for me, not a lot of time to spare. I'll still try to check this problem out in the next few days, but I'm not optimistic about actually managing to. Apologies, it's just a very busy period.

  • That's alright, no worries. I can extend the deadline a bit if you want, but there's absolutely no problem

  • Hello, I took another look at this question and was able to figure out most of the details myself, so there's no need for you to concern about this. Thanks anyway!

  • Ok cool, I'm glad you made it!