General Ring Theory/Principal ideals

Definition (left principal ideal):

Let $R$ be a ring. A left principal ideal of $R$ is an ideal of the form $aR$ with $a\in \mathbb {R}$.

Analogously, one may define a right principal ideal.

Definition (two-sided principal ideal):

Let $R$ be a ring. A two-sided principal ideal is an ideal of the form $RaR$ for some $a\in R$.