Ring with special rules for add and mult

Question

$R$ is a ring of integers with special rules for multiplication and addition. Suppose that $f: \mathbb Z \to R$ is an isomorphism that is defined by $f(x) = x-2$.

1. What integer in the ring $R$ is zero?
2. What integer in the ring $R$ is the identity?
3. What are the units in $R$?
4. Find a formula for the special rule for addition.
5. Find a formula for the special rule for multiplication.

My thoughts: I know rules for the individual elements of the question. My weakness lies with understanding the meaning of Rings and the question in general. Is every element in the ring a product of the function x-2?

Answer 1

An isomorphism preserves the identity elements and units, so the zero in $R$ is $f(0)$ which is $-2$. Questions 2 and 3 are similar.

Denote the addition in $R$ by $x\oplus y$. By definition of isomorphism, $$f(a)\oplus f(b)=f(a+b)\ .$$ If $x,y\in R$ are given then you can work out $a,b\in\Bbb Z$ such that $x=f(a)$, $y=f(b)$, then simplify the preceding expression for $x\oplus y$.

Answer 2

Hints: 1)You know that a ring isomorphism must take $0$ to $0$. So, $0$ in $R$ is just $f(0)$.

2) Does the ring isomorphism take $1$ to $1$? :)

3) Does the ring isomorphism take units to units?

4) $r_1+r_2=f(x)+f(y)$ where $x,y$ are the element in $\mathbb{Z}$ such that $f(x)=r_1, f(y)=r_2$