Comparing left principal ideals in a non commutative ring with unity If $R$ is a non commutative ring with unity. Then $Ra, Rba$ and $Rab$ are left principal ideals for some $a,b\in R$. What is the relationship between these left ideals? For instance, which one of the following is correct? $Rba\subseteq Ra$ or $Rba\subseteq Rb$ All I know is that $Ra=\\{ra:r\in R\\}$ and $Rab=\\{rab:r\in R\\}$.

$ba\in Ra$, therefore $Rba\subseteq Ra$ (it has to absorb on the left.)

There is no reason for $Rba\subseteq Rb$. You would have to show, in particular, that $ba\in Rb$. Without commutativity, this is not a sure thing.

For a counterexample, you can look at $R=\mathbb Z\langle x,y\rangle/(y^2, xy)$, and examine the left ideal $Ry$ and whether or not $xy$ is in it.