Is there a finite number of ways to change the operation in a ring and retain a ring? Consider, for instance, the ring $(\mathbb{Z}, +, \cdot)$.  We can create a new ring $(\mathbb{Z}, \oplus, \odot)$ by defining
$$ a \oplus b = a + b - 1 \quad \text{and} \quad a \odot b = a + b - ab. $$
There are also other operations that we could use as 'addition' and 'multiplication' (for example we could define $a \oplus b = a + b - 1$ and $a \odot b = ab - (a + b) + 2$, etc.)
My question is: is there a finite number of different operations $\oplus$ and $\odot$ that we could define on $\mathbb{Z}$ and retain the ring structure, or are there infinitely many such operations?  what about on $\mathbb{R}$? what about on $\mathbb{Z} / n \mathbb{Z}$?

On $\Bbb {Z/nZ}$ there must be finitely many as there are only finitely many operations on a finite set.
Your examples are derived by considering replacing each element $x$ with $x-1$ and seeing what happens to the operations.  The constant is arbitrary, so you have an infinite number of choices on $\Bbb {Z, R}$
