Let $A$ be a local ring which is also an integral domain and $M$ be a rank one $A$-module. Denote by $k$ the residue field of $A$. Is $\dim M \otimes\_A k \le 1$? If not, is there a known upper-bound for $\dim M \otimes\_A k$?



On rank one torsion-free modules over local rings
$B=k[x\_1,\dots,x\_n]$, 


$\mathfrak m=(x\_1,\dots,x\_n)\vartriangleleft\_{\rm max} B$, 


$A=B\_{\mathfrak m}$,


$M=\mathfrak m\_{\mathfrak m}$.


Then $K:=\mathrm{Frac} A\simeq k(x\_1,\dots,x\_n)\simeq M\otimes\_A K$, so $M$ has rank one, but $\dim\_k M\otimes k=n$.


