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 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 (A lower bound for the canonical height on elliptic curves over abelian extensions, 2003). The higher dimensional case was done by them jointly. 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 $\bigoplus \mathbb{Z}$.

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

Amoroso and Dvornicich 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 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-Dvornicich 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-Dvornicich 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 (A lower bound for the canonical height on elliptic curves over abelian extensions, 2003). The higher dimensional case was done by them jointly. 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 $\bigoplus \mathbb{Z}$.