Let $a$ and $b$ be algebraic numbers which are not necessarily algebraic integers. Is there some invariant that allows us to determine whether $\mathbb Z[a]$ and $\mathbb Z[b]$ are isomorphic as $\mathbb Z$-modules?

Let $a$ and $b$ be algebraic numbers which are not necessarily algebraic integers. Is there some invariant that allows us to determine whether $\mathbb Z[a]$ and $\mathbb Z[b]$ are isomorphic as $\mathbb Z$-modules?

Initial version of the question: Let $a$ be an algebraic number which is not necessarily an algebraic integer. What is the $\mathbb Z$-module structure of $\mathbb{Z}[a]$?

Let a and b be algebraic numbers which are not necessarily algebraic integers. Is there some invariant that allows us to determine Z[a] and Z[b] are isomorphic as Z modules?

If the modified question is still deemed unfit to the community, please close it. Thanks.

Let $a$ and $b$ be algebraic numbers which are not necessarily algebraic integers. Is there some invariant that allows us to determine whether $\mathbb Z[a]$ and $\mathbb Z[b]$ are isomorphic as $\mathbb Z$-modules?