The determinant of transvections (an update)

In the previous post , I had explained how I could prove a general version of the classic fact that transvections have determinant 1. Recall here that transvections in an $R$-module $M$ are linear maps of the form $x\mapsto x+f(x)v$, where $f$ is a linear form on $M$ and $v\in M$ is a vector such that $f(v)=0$. To be able to talk of determinant, I assumed that $M$ had a finite basis, so that…