[DavidSJ]

I think we’re not really disagreeing about anything. In a pure math context, x would be, as you say, a quantity in which dimensions play no role.

[ttoinou]

As he said, x is part of a ring. In physics the multiplication of two elements from the same ring end up on another ring (meters^2 for example) in which, for example, you cannot do addition with elements from the original ring (meters)

[yequalsx]

The mathematical definition of a ring precludes this from happening. By definition, in a ring when you multiply 2 elements from the ring you get another element of the ring. Perhaps you are referring tensor products (geometric algebra for physicists).