Different ring structures on the product of rings Let $(R_1, +_1, \circ_1)$ and $(R_2, +_2, \circ_2) $ be two rings. Let us consider the set $R = R_1 \times R_2$. We know it has a ring structure given by the following:
$$(x,y) + (x^\prime, y^\prime) = (x+_1 x^\prime, y +_2 y^\prime)$$ and $$(x,y)\circ(x^\prime, y^\prime) = (x\circ_1 x^\prime, y \circ_2 y^\prime)$$
Question: I am looking for different ring structures on $R.$ Any help will be appreciated. 
 A: In general if you have a bijective map $f$ of a ring $S$ to itself  then you can define two new operations
$x+_fy:=f^{-1}(f(x)+f(y))$
$x*_fy:=f^{-1}(f(x)f(y))$
In this case $(S,+_f, *_f)$ is a ring, where the neutral element with respect to $+_f$ is $f^{-1}(0)$ and the neutral element with respect to $*_f$ is $f^{-1}(1)$
Example:
If you choose $f: S\to S$ such that 
$f(x):=-x+1$ 
you have that the inverse is $g(x)=-x+1=f(x)$ and in this case we have 
$g(0)=1$ and $g(1)=-1+1=0$ 
So $1$ is the new neutral element with respect to $+_f$ and $0$ is neutral element with respect to $*_f$ on the new ring $(S,+_f,*_f)$. The new operations are
$x+_fy=f(-x-y+2)=x+y-1$
$x*_fy=f((-x+1)(-y+1))=$
$f(xy-x-y+1)=x+y-xy$
The problem is that  $f: (S,+_f,*_f)\to (S,+,*)$ is an isomorphism of rings, so the two structures are equal.
In your case if you want a new structure on $R$, equal to the initial structure up to isomorphism,
you can consider two bijective maps $f:R_1\to R_1$ and $g:R_2\to R_2$, and in this case you have that 
$(R, +_{(f,g)}, *_{(f,g)})$ is a new ring but it is isomorphic to the initial ring $(R,+,*)$ 
If you want a different structure you can consider a generalization of semi-direct products for Rings:
If you have a morphism 
$\psi: R_2\to Aut((R_1,+,*))$
then you have that 
$(a,b)+^\sim(c,d)=(a+\psi(b)(c), b+d)$ 
and 
$(a,b)*^\sim(c,d)=(a\psi(b)(c), bd)$
are two operation on $R_1\times R_2$ such that 
$(R_1\times R_2, +^\sim,*^\sim)$ is a ring different from $(R,+,*)$ 
