Definition (left principal ideal):

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

Definition (two-sided principal ideal):

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