Null Spaces of Sums and Differences of Inner Regular Elements in Unital Rings

In this paper, we investigate the right and left null spaces of sums, differences, and modified sums of inner regular elements in unital rings. We establish sufficient conditions under which these null spaces coincide with intersections of the corresponding kernel ideals, and we derive criteria for their triviality. For commuting {1,5}-invertible elements, we obtain decompositions of the null spaces of differences of associated idempotent-like elements. We also establish equivalence conditions for the invertibility of sums and differences of special classes of generalized invertible elements. Applications to bounded linear operators on Banach and Hilbert spaces, as well as to matrices, are presented.