What are the elements of the modular ring mod 7?
You might be confusing the ring and the multiplicative group of invertible elements mod p.
The first one is isomorphic to the integers from 0 to p-1, 0 being the additive neutral element, and 1 being the multiplicative neutral element.

If p is prime, the intervible elements are the integers from 1 to p-1, and they form a multiplicative group.

By the way, when p is prime, such a ring (all elements except 0 are invertible) is called a field.