What are the elements of the modular ring mod 7? Are the elements of a modular ring simply the set of all the numbers from 1 to p−1? 

in this case p−1 = 6 ?

I asked this on the math stack exchange https://math.stackexchange.com/q/3375667 and was told that 0 is included in the the ring. Is this treated differently in CS ?
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.