Let $\alpha:M_R\to M_R$ be an endomorphism of the right $R$-module $M$ defined by $\alpha(m)=ma$ for some $a\in R$ and for all $m\in M$. Then $$\text{Im}(\alpha)=Ma.$$ If we set $M=R$, then $\alpha:R_R\to R_R$ has $$\text{Im}(\alpha)=Ra.$$ However, I understand that $Ra$ is a left ideal of $R$ and so it is a left $R$-module. My question:
Why is it that the image of $\alpha$, in this case $Ra$, turns out to be a left $R$-module yet I am working with right $R$-modules? I would have expected to have the image as $aR$ which is a right $R$-module.