Generalized Fitting modules and rings

An R-module M is called strongly Hopfian (respectively strongly co-Hopfian) if for every endomorphism f of M the chain Kerf⊆Kerf2⊆⋯ (respectively Imf⊇Imf2⊇⋯) stabilizes. The class of strongly Hopfian (respectively co-Hopfian) modules lies properly between the class of Noetherian (respectively Artinian) and the class of Hopfian (respectively co-Hopfian) modules. For a quasi-projective (respectively quasi-injective) module, M, if M is strongly co-Hopfian (respectively strongly Hopfian) then M is strongly Hopfian (respectively strongly co-Hopfian). As a consequence we obtain a version of Hopkins–Levitzki theorem for strongly co-Hopfian rings. Namely, a strongly co-Hopfian ring is strongly Hopfian. Also we prove that for a commutative ring A, the polynomial ring A[X] is strongly Hopfian if and only if A is strongly Hopfian.