I was computing the left annihilators of the elements $x$ of a ring $R$ with $1_R$ (denoted by Ann$_R(x)$) and encountered the following scenario:
For any $a\in R$, $$\text{Ann}_R(1_R-a)=\{r\in R: r(1_R-a)=0\}=\{r\in R: r1_R-ra=0\}=\{r\in R: r=ra\}.$$ From this observation, I feel like concluding that $$\text{Ann}_R(1_R-a)=Ra.$$
My question is, should I conclude that that $$\text{Ann}_R(1_R-a)=Ra?$$ Does it make sense?
No, all you can conclude is that $r$ is a left-annihilator of $1-a$ if and only if $r=ra$.
That doesn't imply that for all $r\in R$, the element $ra$ is a left-annihilator of $1-a$.
For example, if $a\ne a^2$, the element $1a=a$ is not a left-annihilator of $1-a$.