# Quotients of powers of maximal ideal in a local ring.

I need a reference to a (hopefully) known result:

I think I can quite easily prove that if $L$ is a commutative Noetherian local ring with maximal ideal $M$ then for any positive integer $n$, $L$-module $M^{n-1}/M^n$ is isomorphic to a direct sum $F \oplus F \oplus \dots \oplus F$ of finite number of instances of field $F = L/M$. Moreover, the number of copies of $F$ equals the number of generators of $M^{n-1}$.

It would be very surprising if such result would not have been previously known. I would appreciate any pointers. Thank you very much!

This is a more general fact about finitely generated $$L$$-modules. If $$X$$ is a finitely generated $$L$$-modules, then $$X/MX$$ is a direct sum of a finite number of copies of $$F$$, and the number of such copies is the same as the cardinality of any minimal generating set for $$X$$. Your result is then just the special case $$X=M^n$$.
I don't know a specific reference for this exact statement, but it is a well-known corollary of Nakayama's lemma (and I would consider it reasonable to just state that it is true "by Nakayama's lemma" without needing any more specific reference). The fact that $$X/MX$$ is a direct sum of copies of $$F$$ is trivial: $$X/MX$$ is a module over the ring $$L/M=F$$, but $$F$$ is a field so any $$F$$-module (i.e., vector space) is a direct sum of copies of $$F$$. Then, by Nakayama's lemma a subset of $$X$$ generates it iff its image in $$X/MX$$ generates $$X/MX$$. So, a minimal generating set for $$X$$ has the same cardinality as a minimal generating set for $$X/MX$$, which is just the dimension of $$X/MX$$ as a vector space over $$F$$.