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Sibirsk. Mat. Zh., 2019, Volume 60, Number 3, Pages 630–639 (Mi smj3099)  

Decompositions of dual automorphism invariant modules over semiperfect rings

Y. Kuratomi

Department of Mathematics, Faculty of Science, Yamaguchi University, Yoshida, Yamaguchi, Japan

Abstract: A module $M$ is called dual automorphism invariant if whenever $X_1$ and $X_2$ are small submodules of $M$, then each epimorphism $f: M/X_1\to M/X_2$ lifts to an endomorphism $g$ of $M$. A module $M$ is said to be $\mathrm{d}$-square free (dual square free) if whenever some factor module of $M$ is isomorphic to $N^2$ 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 $\mathrm{d}$-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.

Keywords: dual automorphism invariant module, pseudoprojective module, dual square free module, finite internal exchange property, (semi)perfect ring.

Funding Agency Grant Number
Japan Society for the Promotion of Science 15K04821
This work was supported by JSPS KAKENHI Grant Number 15K04821.


DOI: https://doi.org/10.33048/smzh.2019.60.311

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English version:
Siberian Mathematical Journal, 2019, 60:3, 490–496

Bibliographic databases:

UDC: 512.55
MSC: 35R30
Received: 19.07.2018
Revised: 15.11.2018
Accepted:19.12.2018

Citation: Y. Kuratomi, “Decompositions of dual automorphism invariant modules over semiperfect rings”, Sibirsk. Mat. Zh., 60:3 (2019), 630–639; Siberian Math. J., 60:3 (2019), 490–496

Citation in format AMSBIB
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\paper Decompositions of dual automorphism invariant modules over semiperfect rings
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\vol 60
\issue 3
\pages 630--639
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\crossref{https://doi.org/10.33048/smzh.2019.60.311}
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\transl
\jour Siberian Math. J.
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\pages 490--496
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