Let $R$ be a ring with unity and $a,b\in R$. For a right $R$ module $M$, define $\text{Ann}^M(r)=\{m\in M:mr=0\}$ as a right $R$-module:
My question is, does the isomorphism $\text{Ann}^M(1-ab)\cong \left(\text{Ann}^M(1-ab)\right)a$ always hold?
I believe it holds if $a$ is a unit element in $R$. My attempt for the proof is: Define two morphisms $f:\text{Ann}^M(1-ab)\to \left(\text{Ann}^M(1-ab)\right)a$ by $f(m)=ma$ and $g:\left(\text{Ann}^M(1-ab)\right)a\to \text{Ann}^M(1-ab)$ by $g(ma)=m$. My deduction is $fg=1$ and $gf=1$ and so $f$ is an isomorphism. However, I am not sure whether the isomorphism exists.