Proving that structure is a ring. Lets say, I have a set S with two operations defined in it: + and *. I need to prove that this structure is a ring.
I also have a structure R, which i know is a ring, and a function:
$$f: S \rightarrow R $$
Is it enough to prove that f is isomorphism between S and R?

If you can show that $f$ is a bijection that preserves the operations, you will have shown that $S$ is a ring.
To see why this is true, let's check distributivity of addition: we have, for any $a,b,c\in S$,
$$\begin{align}f\big(a*_S(b+_Sc)\big) &= f(a) *_R\big(f(b)+_Rf(c )\big) &&\text{(since $f$ is surjective and preserves}\\&&&\text{ring operations)} \\&= \big(f(a)*_Rf(b)\big) +_R\big(f(a)*_Rf(c )\big)&&\text{(by distributivity in $R$)}\\&=f\big((a*_Sb )+_S(a*_Sc)\big)
\end{align}$$
Since $f$ is an injection, it follows that $$a*_S(b+_Sc)=(a*_Sb )+_S(a*_Sc)$$
The other properties follow similarly.
In fact, provided that you know that $S$ is closed under addition and multiplication, which is implicit in saying that $*$ and $+$ are defined and that you know $S$ has a multiplicative and additive identity, then you only need $f$ to be injective.
