set theory - Is it possible to decrease the rank of known structures?

Recall that for a set $x$ its rank $alpha$ is the least ordinal such that $x in V_{alpha+1}$. Or in other words: $x$ is built up out of $alpha$ levels of braces and the empty set.

I think with the usual constructions of numbers (cartesian products, sets of equivalence classes, Dedekind cuts, etc.), we have

$rank(mathbb{N})=omega, rank(mathbb{Z})=omega+4, rank(mathbb{Q})=omega+8, rank(mathbb{R})=omega+10$

Now it is possible to find a bijection $mathbb{Z} cong mathbb{N}$, so that there is a copy of $mathbb{Z}$ of smaller rank, namely $omega$. But this is, of course, nonsense. We should also consider the ring structure on $mathbb{Z}$, which is given by two maps $mathbb{Z} times mathbb{Z} to mathbb{Z}$. Therefore we might ask the following:

Is there a ring $(R,+,*)$, which is isomorphic to $(mathbb{Z},+,*)$, but $rank(R,+,*) < rank(mathbb{Z},+,*)$? What about the other rings above?

Of course, this question is just out of curiosity. I doubt that anybody cares about these bounds of ranks (if not, please let me know).