Abstract
A module M is called dual automorphism invariant if whenever X1 and X2 are small submodules of M, then each epimorphism f : M/X1 → M/X2 lifts to an endomorphism g of M. A module M is said to be d-square free (dual square free) if whenever some factor module of M is isomorphic to N2 for a module N then N = 0. We show that each dual automorphism invariant module over a semiperfect ring which is a small epimorphic image of a projective lifting module is a direct sum of cyclic indecomposable d-square free modules. Moreover, we prove that for each module M over a semiperfect ring which is a small epimorphic image of a projective lifting module (e.g., M is a finitely generated module), M is dual automorphism invariant iff M is pseudoprojective. Also, we give the necessary and sufficient conditions for a dual automorphism invariant module over a right perfect ring to be quasiprojective.
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Russian Text © The Author(s), 2019, published in Sibirskii Matematicheskii Zhurnal, 2019, Vol. 60, No. 3, pp. 630–639.
The author was supported by the JSPS KAKENHI Grant Number 15K04821.
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Kuratomi, Y. Decompositions of Dual Automorphism Invariant Modules over Semiperfect Rings. Sib Math J 60, 490–496 (2019). https://doi.org/10.1134/S003744661903011X
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DOI: https://doi.org/10.1134/S003744661903011X