Ring whose all ideals are double-sided is commutative?

Question

I was thinking about the following problem:

Suppose R is a ring s.t. every left ideal is also right. Is R commutative?

This actually continues the easier question:

Suppose G is a group whose all subgroups are normal. Is G commutative?

The answer to the latter is negative, since one can take the quaternion group of order 8.

Edit: One can indeed take a division ring as an counterexample. I am looking for something more interesting, in particular with some proper ideals.

Thanks

Answer 1

Not sure if this is satisfactory, but division rings only have trivial left and right ideals.

Therefore finite products of noncommutative division rings are counterexamples.

Answer 2

A ring is called a left duo ring if every left ideal is also a right ideal. (It's called a duo ring if every one-sided ideal is two-sided.) There are many rings which are duo on one side but not commutative.

As already discussed, division rings and finite products of them are duo rings.

But putting them aside, you could also make a duo ring by taking your favorite noncommutative division ring $D$ and using $D[x]/(x^2)$. It has exactly three ideals of any type (the trivial ones and $(x)$) and is not commutative.