Magnetized vortex in three-dimensional

Modified Gravity Theories (MGTs) are extensions of General Relativity (GR) in its standard formulation. Therefore, within this framework, we will investigate a system composed of a black hole (BH) surrounded by Maxwell-Higgs vortices, forming the BH-vortex system. In the case of linear f(R) gravity it is adopted showing the existence of a three-dimensional ring-like BH-vortex system with quantized magnetic flux. Within this system, one notes the BH at r = 0 and its event horizon at $r= r_{0}$ , while the magnetic vortices are at $r \in (r_{0}, \infty )$ . A remarkable result is the constancy of the Bekenstein-Hawking temperature (TH), regardless of MGTs and vortex parameters. This invariance of TH suggests that the BH-vortex system reaches thermodynamic stability. Unlike the standard theory of Maxwell-Higgs vortices in flat spacetime, in f(R) gravity, the vortices suffer the influence of the BH's event horizon. This interaction induces perturbations in the magnetic vortex profile, forming cosmological ring-like magnetic structures.