Isomorphisms of rings and their generators It's been a while since I touched abstract algebra, so please correct me if I'm wrong here: if I wanted to construct an isomorphism between the polynomial rings with integer coefficients $\mathbb{Z}[x_1+x_2]\cong \mathbb{Z}[x_1+x_1^{-1}+1]$, I would identify $x_1$ with $x_1$ and $x_2$ with $x_1^{-1}+1$; would the explicit isomorphism $\varphi:\mathbb{Z}[x_1+x_2]\to \mathbb{Z}[x_1+x_1^{-1}+1]$ be simply to take the generator $\varphi(x_1+x_2)=x_1+x_1^{-1}+1$?
Also, please bear with me a bit, but if I wanted an explicit isomorphism
$\mathbb{Z}[x_1+ x_1^{-1}+x_2+x_2^{-1}+1,2+x_1x_2+x_1x_2^{-1}+x_1+x_1^{-1}x_2+x_1^{-1}x_2^{-1}+x_1^{-1}+x_2+x_2^{-1}]\cong \mathbb{Z}[x_1+x_1^{-1}+x_2+x_2^{-1},1+x_1x_2+x_1x_2^{-1}+x_1^{-1}x_2+x_1^{-1}x_2^{-1}]$,
would I just send the generators to each other, e.g. $x_1+ x_1^{-1}+x_2+x_2^{-1}+1\mapsto x_1+x_1^{-1}+x_2+x_2^{-1}$ and $2+x_1x_2+x_1x_2^{-1}+x_1+x_1^{-1}x_2+x_1^{-1}x_2^{-1}+x_1^{-1}+x_2+x_2^{-1}\mapsto $
$1+x_1x_2+x_1x_2^{-1}+x_1^{-1}x_2+x_1^{-1}x_2^{-1}$?
Is there perhaps another way to realize this isomorphism?
 A: I think you'll have more luck here if you're explicit about what these rings are.  For instance, what is $\mathbb{Z}[x_1+x_2]$?  Polynomial rings have very concrete definitions.  So you should try to realize all of your rings as subquotients (i.e. quotients of subrings) of polynomial rings.  For instance, $\mathbb{Z}[x_1+x_2]$ is the subring of the polynomial ring $\mathbb{Z}[x_1,x_2]$ generated by $x_1+x_2$.  Similarly, $\mathbb{Z}[x_1+x_1^{-1}+1]$ is the subring of $\mathbb{Z}[x_1,x_1^{-1}]$ generated by $x_1+x_1^{-1}+1$.  One can define $\mathbb{Z}[x_1,x_1^{-1}]$ to be the quotient of the polynomial ring $\mathbb{Z}[x_1,y_1]$ by the ideal generated by $x_1y_1-1$ (so that the image of $y_1$ in the quotient corresponds to $x_1^{-1}$).
Once you've defined your rings precisely, you can then use the fact that one can define a map from a polynomial ring (over $\mathbb{Z}$) to any commutative ring by mapping the indeterminates to any elements you like (this is a very nice property of polynomial rings).  To define a map from a quotient of a polynomial ring to another commutative ring, you can map the indeterminates to any elements you like, provided that the ideal you've quotiented out by is mapped to zero.
To apply this concretely to your first question, there is a map from $\mathbb{Z}[x_1,x_2]$ to $\mathbb{Z}[x_1,x_1^{-1}]$ that sends $x_1$ to $x_1$ and $x_2$ to $x_1^{-1}$.  Restricting this map to the subring $\mathbb{Z}[x_1+x_2] \subseteq \mathbb{Z}[x_1,x_2]$ yields a map $\mathbb{Z}[x_1+x_2] \to \mathbb{Z}[x_1,x_1^{-1}]$.  Since the image of this map lies in $\mathbb{Z}[x_1 + x_1^{-1}+1]$ (which, incidentally, is equal to $\mathbb{Z}[x_1+x_1^{-1}]$), you can restrict the codomain to get the map you want.  This gives exactly that map you said: it maps $x_1+x_2$ to $x_1 +x_1^{-1} + 1$, but it justifies why such a map exists.
Now try the same thing with your second map.  This will be little bit more involved, but the idea is the same.
