Exercise
Suppose that R is a ring. Which of the following statements is true?
If R is commutative, then so is R*.
If R* is commutative, then so is R.
Of course,
the multiplication of R* comes from the commutative multiplication
* of R.
No. Here is a somewhat advanced example.
Let R be the ring of all "polynomials" in the two noncommuting variables X and Y with coefficients in Q. Then clearly R is not commutative. But the only invertible elements in R are the nonzero elements in Q (the constant polynomials).