Sure, the integers form a ring under the standard operations of addition and multiplication.
But you can also form a trivial ring by defining a*b=0 for all a,b. It is kind of lame, but it is still a ring. With these two types of multiplication imposed on the integers it is clear there is no ring isomorphism between the two.
The standard integer operations have a multiplicative identity, 1. Unity. This other multiplication clearly has no unity because for any nonzero element a.
for all b, so there is no identity for the nonzero integers and the unity must give back the element for multiplication on everything in the ring.