Background and Objectives: One of the most promising properties of thin quantum rings – their selective properties for localized electrons in a magnetic field – is practically not discussed in the modern literature. Meanwhile, the spectrum of such rings can be reduced to a single stable level, all quantum numbers of which (including spin number) are controlled by the magnetic field. The electronic states of flat thin quantum rings of rectangular cross section, whose thickness h, inner radius R_in and outer radius R_ex are related by the relations h ≪ R_ex − R_in ≪ R_in, are considered; hereinafter we will call them “quantum shims”. This paper is devoted to select the parameters of a quantum shim so that it holds an electron only in a particular spin state. Materials and Methods: The equations for the radial function, which are of primary physical interest, are obtained by analytically solving the Schrödinger equation for this structure. Results: It has turned out that narrow-gap quantum shims in a wide-gap matrix have the most pronounced and fully controllable selection properties. To keep an electron in such a shim at a single stable level, an external homogeneous magnetic field of strictly defined strength is required. Each set of quantum numbers – radial, orbital and spin – corresponds to a unique value of field strength stabilizing this level. Conclusion: It has been found that this type of narrow-gap heterostructures in a wide-gap matrix can become basic elements for spintronic systems. Their spectrum in an external magnetic field can be reduced to a single stable level, all quantum numbers of which (spin including) are controlled by the external field. We have considered variants of changing the spin state of an electron localized on a shim by a longitudinal magnetic field.