Since you ask more generally for results on $E(\mathbb{Q}^{\mathrm{ab}})$, let me expand my comment into a short answer. 

Amoroso and Dvornicic discovered (*A lower bound on the height in abelian extensions*, JNT 2000) that the absolute logarithmic height on $\mathbb{G}_m(\bar{\mathbb{Q}})$ is bounded below by the absolute positive constant $(\log{5}) /12$ on the subset $\mathbb{G}_m(\mathbb{Q}^{ab}) \setminus \mu_{\infty}$ of abelian points minus the torsion. A simplified proof of this result (with a weakened constant) is in chapter four of Bombieri and Gubler's *Heights in Diophantine Geometry*. The problem of finding the best possible constants (the infimum and the limit infimum of the canonical height on abelian non-torsion points) has remained unsolved. 

Lawrence had [proved][1] in 1984 that a countable abelian group with a discrete norm is isomorphic to the direct sum $ \bigoplus \mathbb{Z}$ of copies of $\mathbb{Z}$. Since Amoroso-Dvornicic implies that the canonical height descends to a discrete norm on the abelian group $\mathbb{G}_m(\mathbb{Q}^{\mathrm{ab}}) / \mu_{\infty}$, this recovers Iwasawa's old result about the freeness of $(\mathbb{Q}^{\mathrm{ab}})^{\times} / \mathrm{tors}$. 

The elliptic analog of the Amoroso-Dvornicic theorem was proved, following the original $\mathbb{G}_m$ blueprint, by Baker (*Lower bounds for the canonical height on elliptic curves over abelian extensions*, 2002) and [Silverman][2] (*A lower bound for the canonical height on elliptic curves over abelian extensions*, 2003). The higher dimensional case [was done by them jointly][3]. The proof of this result also includes Ribet's result you cited as a particular case.

In particular, since this implies again that the canonical height descends to a discrete norm on the abelian group $E(\mathbb{Q}^{\mathrm{ab}}) / \mathrm{tors}$, it follows again that the latter group is isomorphic to $\bigoplus \mathbb{Z}$. 


  [1]: https://www.ams.org/journals/proc/1984-090-03/S0002-9939-1984-0728346-9/S0002-9939-1984-0728346-9.pdf
  [2]: https://arxiv.org/pdf/math/0305041.pdf
  [3]: https://arxiv.org/pdf/math/0312393v1.pdf