Why principal ideal should be commutative? According to the definition of Principal Ideal it should be commutative. What if the ring is not commutative? Which means $ar\neq ra$ where $a\in I, r \in R$. Does it lead to a contradiction? Because I can't find a contradiction.
 A: For noncommutative rings there are three different notions of ideal:


*

*left ideal (absorbs ambient multiplication from the left)

*right ideal (absorbs ambient multiplication from the right)

*two-sided ideal (absorbs ambient multiplication from both sides)


A left/right/two-sided ideal is principal if it is the smallest such ideal containing a given element of the ring (we say that element generates the ideal). One can show that $Ra$ and $aR$ are respectively the left and right ideals principally generated by an $a\in R$, where $Ra:=\{ra:r\in R\}$ and then similarly $aR:=\{ar:r\in R\}$. The two-sided principal ideal is more complicated to describe.
It seems you are asking if $aR=Ra$ always holds, even if $R$ is noncommutative. The answer is no it doesn't. In fact, none of the left, right and two-sided ideals principally generated by a single element are necessarily the same. Furthermore, the right ideal $aR$ generally fails to be a left ideal, and then symmetrically the left ideal $Ra$ may fail to be a right ideal, and further the two-sided ideal $(a)$ when considered as a left or right ideal may fail to be principal.
For examples of where this (pathological, one may at first feel) behavior, consider forming the noncommutative polynomial ring $k\langle a,b,\cdots\rangle$ out of some letters $a,b,\cdots$ and a nice ring $k$.
