The spectral rank of an element in a ring

The concept of the spectral rank of an element in a ring is defined, and it is shown to be a genuine generalization of the same concept first studied in the setting of a Banach algebra. Furthermore, we prove that many of the desirable properties of this rank are still valid in the more abstract setting and give several examples to support and motivate the given definition. In particular, we are able to show that a nonzero idempotent of a semiprimitive and additively torsion-free ring is minimal if and only if it has a spectral rank of one. We also discover a precise connection between the spectral rank of an element in a ring and a purely algebraic definition of rank considered only recently by N. Stopar in [Rank of elements of general rings in connection with unit-regularity, J. Pure Appl. Algebra 224 (2020) 106211]. Specifically, we are able to show that if an elementa has a finite algebraic rank in a semiprimitive and additively torsion-free ring, thena has the exact same spectral rank. An extra condition under which the converse holds true is also provided, and connections to the socle are identified. Finally, for both of these extended notions of rank considered in the setting of a ring, we prove a generalized Frobenius Inequality.