Cyclic left $R$-module $Ra$ is projective if and only if $\mathrm{Ann}_R(a)$ is of the form $Re$ for some $e \in R $ with $e^2=e$

Question

Let $R$ be a ring with unity. For a nonzero $a\in R$, $\mathrm{Ann}_R(a)= \{r \in R \mid ra=0\}$.

How to show that the cyclic left $R$-module $Ra$ is projective if and only if $\mathrm{Ann}_R(a)$ is of the form $Re$ for some $e \in R $ with $e^2=e$ ?

Answer 1

$\newcommand\Ann{\operatorname{Ann}}$Recall that $Ra\cong R/\Ann_R(a)$, thus $Ra\cong R/Re$ in our situation.

If $e^2=e$ and $q:Y\to R/Re$ is a surjective homomorphism, then $x+Re\mapsto x(y-ey):R/Re\to Y$ is a section of $q$ if $q(y)=1+Re$. This proves $R/Re$ projective.

Conversely, if $Ra$ is projective, the surjection $x\mapsto xa:R\to Ra$ has a section $s:Ra\to R$. Since $s(a)a=a$, we have $s(a)=1-e$ for some $e\in\Ann_R(a)$. For every $r\in\Ann_R(a)$, we have $0=s(ra)=r-re$ thus $r=re$. In particular, $e^2=e$ and $\Ann_R(a)=Re$ which concludes the proof.