Left Ideal right ideal identity

Question

If the left ideal equals the right ideal (denoted as $I$) in a ring $R$ does that mean $rI = Ir = I$ for any $r$ inside $R$?

Also, in general, if left ideal is the same as right ideal in a ring, does that mean the ring is commutative? (I think that if the ring is commutative, then any right ideal is the same as left ideal)

Answer 1

A ring is called right duo If all of its right ideals are also left ideals.

There exist one-sided duo rings

A right-and-left duo ring does not have to be commutative. Example: the quaternions.

if $I$ is a left and right ideal does $Ir=I=rI$ for all r ?

No, it only means $rI\subseteq I$ and $Ir\subseteq I$. As mentioned in the comments, any nonzero ideal of $\mathbb Z$ will be an example for you.