Isomorphic groups but not isomorphic rings Provide an example of two rings that have the same characteristic, are isomorphic as groups but are not isomorphic as rings.
I'm confused with how to being. I know that having the same characteristic means that the concatenation is the same number to receive the zero element.

The group isomorphism refers to the additive structure.
Let $R$ be any ring. We can define two ring structures on the set $R\times R$: the addition is the same, so the two additive groups are not only isomorphic, but identical:
$$
(a,b)+(c,d)=(a+c,b+d)
$$
We can define two different multiplications:
$$
(a,b)\cdot(c,d)=(ac,bd)
$$
and
$$
(a,b)*(c,d)=(ac,ad+bc)
$$
It's not difficult to show that $(R\times R,+,\cdot)$ and $(R\times R,+,*)$ are rings. Can you find the characteristic of them and a case where the two rings are not isomorphic?
