The electronic states of flat thin quantum rings of rectangular cross section whose thickness h, inner radius R_in and outer R_ex are related by the relations ; in are considered; hereinafter we will call them “quantum shims”. It is 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. This is proved both by numerical calculations and approximate analytical estimates, clarifying the mechanism of formation of such a state. Because of the conservation of an undamped quantum current and the associated magnetic moment, these states are more stable than in ideal quantum dots with a similar spectrum. The variants of changing the spin state of the electron localized on the puck by a longitudinal magnetic field are considered