Isomorphic groups but not isomorphic rings

$\mathbb{R} \times \mathbb{R}$ and $\mathbb{C}$ are isomorphic as additive abelian groups but they have a different multiplicative structure

One easy example is to take a prime $p$ and consider the finite field $\mathbb{F}_{p^2}$ of order $p^2$ and the direct product ring $\mathbb{F}_p \times \mathbb{F}_p$. Both rings have characteristic $p$. As groups, $\mathbb{F}_{p^2}$ and $\mathbb{F}_p \times \mathbb{F}_p$ are both isomorphic to $\mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z}$, but they are not isomorphic as rings because the former is a field while the latter is not.

Somewhat surprisingly, but assuming the axiom of choice $\Bbb R$ and $\Bbb C$ make such example.

Both are $\Bbb Q$-vector spaces of the same dimension, therefore they are isomorphic as $\Bbb Q$-vector spaces, and in particular as groups. However, they are clearly not isomorphic as rings. The same arguments also works with $\Bbb R^n$, and even spaces like $\ell_\infty$ and $\ell_2$.

Interestingly, the existence of such isomorphism(s) relies heavily on the existence of a Hamel basis, and we can prove that it is possible to have a world where the axiom of choice fails, and $\Bbb R$ and $\Bbb C$ are not isomorphic.

In the same spirit, but without relying on the axiom of choice, you can take the [canonical] algebraic closure of $\Bbb Q$, $\overline{\Bbb Q}$, and $\Bbb Q$'s real-closure which is $\overline{\Bbb Q}\cap\Bbb R$. Both are $\Bbb Q$-vector spaces of the same dimension, in this case $\aleph_0$, so they are isomorphic. But they are clearly not isomorphic as fields, since only one of them has a root for $-1$. You can replace one of the fields with any countable but infinite extension of $\Bbb Q$, like $\Bbb Q(\pi)$ for example.

In this case the axiom of choice is avoided, since everything is countable and we can prove the existence of a basis for these vector spaces from this fact alone.

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?