$R$ is a ring of even integers with special rules for multiplication and addition. Suppose that $f : Z \to R$ is an isomorphism that is defined by $f(x) = 2x+4$. What are the special rules for addition and multiplication?
I am given a hint that i should use $a = f(x)$ and $b=f(y)$. Because it is an isomorphism I know that $f(x)+f(y) = f(x+y)$. And in the end i need it to be in terms of $a$ and $b$.
On
Hint: what element of $R$ corresponds to $0$ in $\Bbb Z$? what element corresponds to $1$? I would give them hats to keep from confusing them with elements of $\Bbb Z$. Now since you have $0+0=0$ you need $f(0) \oplus f(0)=f(0)$ and the others. Play with them and you can define $\oplus, \otimes$
Take any elements $a$ and $b$ in R, and solve $f(x)=a$ and $f(y)=b$ to find $x$ and $y$ in terms of $a$ and $b$. Then $x$ and $y$ are in $\mathbb{Z}$ so add them and multiply them the normal ways to find $x + y$ and $x y$ (again, in terms of $a$ and $b$). Then use $f$ to compute the image of those combinations back in R.