Does the characteristic of a ring R without zero divisors 0 or prime? [duplicate]

This is in connection with the question: Showing that the characteristic of a commutative ring R without zero divisors is 0 or prime.

I have found a solution without using commutativeness. I don't understand why commutativeness will be necessary to prove it.

Here is my solution:

If possible let char $$R = pq$$ where $$p,q\in\mathbb N$$ such that $$p,q>1.$$

There exists $$a,b\in R$$ such that $$pa\ne0,qb\ne0.$$

Then $$(pa)(qb)\ne0,$$ a contradiction since $$(pa)(qb)=(pq)(ab)=0.$$

Thus char $$R = 0$$ or prime.

Please tell me where did I go wrong.