3
votes
1answer
85 views

Equivalence of definitions of Krull dimension of a module

I've seen two definitions of Krull dimension of a module $M$ over a (commutative) ring $R$, and their equivlance does not seem obvious: Matsumura on page 31 of his book Commutative Ring Theory ...
2
votes
0answers
68 views

Defining the Rank of a Projective Module

I am trying to understand the definition of rank for a projective module over a noncommutative ring. The definition I am using is: A sufficient condition for the rank of a free module over a ring ...
2
votes
1answer
163 views

On Krull dimension of $M/(0 :_{M} \mathfrak{m}^t)$ module

Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring and $M$ is an $R$-module. There is an non-negative integer $t$ such that $M/(0 :_{M} \mathfrak{m}^t)$ is finitely generated. Then $$\dim ...
3
votes
1answer
184 views

A detail in the proof of Auslander-Buchsbaum Theorem

I'm trying to understand the proof of a theorem (Auslander-Buchsbaum) which says that given a local ring $(R,m,K)$, where $m$ is the maximal ideal and $K$ the residue field, and a finitely generated ...