Multiplying a ring(or an ideal) by an element.

We know $aR$ is an ideal because for $r,r^\prime\in R,x\in aR$ we have $ar+ar^\prime=a(r+r^\prime)\in aR$ and $(ar)b=a(rb)\in aR$ (the proof for left ideals is similar).

We don't necessarily expect $r\cdot R$ to be equal to $R$, in fact, if it is then the ideal is trivial. It is precisely when $r\cdot R\neq R$ that things begin to become interesting. Fields, because every (nonzero) element is invertible, do not have any interesting ideals.