I'm looking for two rings that have the same additive group, but the multiplication is defined differently. For instance, if we have $(\mathbb{Z}_6,+)$ as the additive group, we naturally have the ring $(\mathbb{Z}_6,+,\cdot)$. But can we define multiplication differently so that it still satisfies the ring axioms with some other operation? This new ring, call it $(\mathbb{Z}_6,+,*)$ should not be isomorphic to $(\mathbb{Z}_6,+,\cdot)$.
I found two trivial examples. The identity ring works out, but again, it's trivial. The other one I found was the opposite ring, which in the non-Abelian case is just defined as switching the order of a given ring. Except that's isomorphic to the original ring, so it's not really what I'm looking for.
Are there any two rings that satisfy this property? Any finite rings? Given an arbitrary ring, can we find another non-isomorphic ring with the same group, or does the first ring need to satisfy specific properties to allow this?