Abstract Let $${{\mathcal {S}}}{{\mathcal {S}}}$$ S and {C}}}$$ C be the strictly singular cosingular operators acting between Banach spaces, let $$P\Phi _+$$ P Φ + perturbation classes for upper ...

‎A module $M$ is lifting if and only if $M$ is amply supplemented and‎ ‎every coclosed submodule of $M$ is a direct summand‎. ‎In this paper‎, ‎we are‎ ‎interested in a generalization of lifting modules by removing the condition‎"‎amply supplemented‎" ‎and just focus on modules such that every non-cosingular‎ ‎submodule of them is a summand‎. ‎We call these modules NS‎. ‎We investigate some gen...

An R-module M is called strongly noncosingular if it has no nonzero Rad-small (cosingular) homomorphic image in the sense of Harada. It is proven that (1) an R-module M is strongly noncosingular if and only if M is coatomic and noncosingular; (2) a right perfect ring R is Artinian hereditary serial if and only if the class of injective modules coincides with the class of (strongly) noncosingula...

Certain norm-related functions have been considered in order to obtain characterisations and perturbation results for various classes of semi-Fredholm-type operators. In the §2, by means of what we have called a perturbation function, we present a general approach to the question of obtaining characterisations and perturbation theorems for F + and strictly singular operators. To this end, for e...