The Protasov-Zelenyuk topology and ideal convergence

The so-called $T$-sequences $\mathbf u$ in a group $G$, and the related finest Hausdorff group topology $T_\mathbf u$ on $G$ that makes $\mathbf u$ a null sequence, were introduced by Protasov and Zelenyuk 35 years ago and since then they became a fundamental tool in the field of topological groups. More recently, in the abelian case, the subfamily of $T$-sequences called $TB$-sequences was introduced, as well as the finest precompact group topology $T_\mathbf{pu}$ that makes $\mathbf u$ a null sequence. Here we study the counterpart of all these notions with respect to ideal convergence in place of the classical notion of convergence of a sequence. Also, we study their relation to the already established field of $I$-characterized subgroups of compact abelian groups.