Let $(R,+,\times,0,1)$ be an unitary ring. I need to find two new operations such that $(R,+_1,\times _1,1)$ is also a ring with the new zero element the $1$ of the first ring. I've thought of the new sum as:
$$a+_1b:=a+b-1$$
so that we have:
$$a+_1 1= a+1-1=a$$
But I don't see how to conclude, any idea?
Hint: Take a bijection $\phi:R\rightarrow R$. Since $(R,+,0)$ forms an abelian group, $R$ forms an abelian group under the operation $$s\oplus t := \phi^{-1}(\phi(s)+\phi(t)),\quad s,t\in R,$$ where $e=\phi^{-1}(0)$ is the zero element.