Arbitrary Ring and Field Contruction Does anybody know if there has been any study in the construction of rings and/or fields out of groups, particularly with respect to certain operations?  For example, the set of the reals and operation $a \cdot b := a + b - 1$ forms a commutative group, but how would one find a second operation that would create a field or ring? If it exist, could you construct it out of addition, multiplication and their inverses? How would you approach problems like these in general? 

In general when you have a structure $M$ (in some language L; in this case L is the language or ring and the L-structure is $\mathbb{R}$) and a set $S$ such that there exists a bijective map $f:M\to S$ then you can pass the structure of $M$ to the set $S$ and you get a new $L$-structure. 
In the case of the Rings you can define:
$+^\sim : S\times S\to S$ such that for every $(a,b)\in S\times S$ $a+^\sim b:=f(f^{-1}(a)+f^{-1}(b))$
$\cdot^\sim : S\times S\to S$ such that for every $(a,b)\in S\times S$  $a\cdot^\sim b:=f(f^{-1}(a)f^{-1}(b))$
So you can prove that $(S,+^\sim, \cdot^\sim, f(0), f(1))$ is Ring (with unity if $M$ is a ring with unity).
Now we can define a trivial bijective map to $f:\mathbb{R}\to \mathbb{R} $ , the function that maps every $a\in \mathbb{R}$ to $f(a):=a+1$ 
In this case $ \mathbb{R}$ is a new Ring with unity with the new operations:
$+^\sim : \mathbb{R} \times \mathbb{R}\to \mathbb{R} $ such that for every $(a,b)\in \mathbb{R}\times \mathbb{R} $  $a+^\sim b:=f(f^{-1}(a)+f^{-1}(b))=$
$f((a-1)+(b-1))=f((a+b)-2)=a+b-1$
$\cdot^\sim :  \mathbb{R}\times  \mathbb{R}\to  \mathbb{R} $ such that for every $(a,b)\in  \mathbb{R}\times  \mathbb{R} $  $a\cdot^\sim b:=$
$f(f^{-1}(a)f^{-1}(b))=f((a-1)(b-1))=$
$f(ab-(a+b)+1)=ab-(a+b)+2$
You can observe that $f(0)=1$ so the neutral element is changed with the neutral element with the product (but $f(1)=2$ so 0 it is not the new neutral element with respect to the new product) 
You can find a bijective map that map $f(0)=1$ and $f(1)=0$ in general for Rings.
What is this map? 
