Extending and Lifting of Endomorphisms and Automorphisms of Modules over Non-Primitive HNP Rings

This paper is a review of results on endomorphism-extendable modules, automorphism-extendable modules, pi-injective modules, endomorphism-liftable modules, and automorphism-liftable modules over non-primitive hereditary Noetherian prime rings. A module M is said to be endomorphism-extendable (resp., automorphism-extendable) if every endomorphism (resp., automorphism) of any submodule of the module M can be extended to an endomorphism of the module M, i.e., for any epimorphism h: M -> M and every endomorphism (f) over bar of the module (M) over bar, there exists an endomorphism f of the module M with (f) over barh= hf. A module M is said to be endomorphism-liftable (resp., automorphism-liftable) if every endomorphism (resp., automorphism) of any homomorphic image of the module M can be lifted to an endomorphism of the module M, i.e., for any epimorphism h: M -> (M) over bar and every endomorphism (f) over bar of the module (M) over bar, there exists an endomorphism f of the module M with fh = hf. In particular, a module M over a non-primitive Dedekind prime ring R is an idempotent-liftable, automorphism-liftable module if and only if M is an endomorphism-liftable module if and only if one of the following conditions holds: 1) M is a singular module and every primary component M(A) of the module M is either an indecomposable injective module or a projective RI r(M(A))-module; 2) M = T circle plus F, where T is a singular injective module such that all primary components are indecomposable and F is a finitely generated projective module; 3) M is projective; 4) there are two positive integers k and n such that R is the matrix ring DTh , where D is a complete local Dedekind domain (not necessarily commutative), M = M-1 circle times M-2, M-1 is a projective module of finite rank, and the R-module M-2 is isomorphic to E-k, E is a minimal right ideal of the classical ring of fractions of R. If R is a non-primitive hereditary Noetherian prime ring and M is a singular right R-module, then M is automorphism-liftable if and only if every A-primary component M(P) of the module M is either a projective R/r(M(A))-module or a uniserial injective module M(A) such that all proper submodules are cyclic and form a countable chain 0 = x(0) R subset of x(1) R subset of ..., all subsequent factors of this chain are simple modules and there exists a positive integer n such that x(k) R congruent to x(k+n)/x(n)R and M(A)/x(k) R congruent to M(A)/x(k+n) R for all k = 0, 1, 2, .... Some remarks and open questions are given.