A study on dual square free modules
Algebra and discrete mathematics, Tome 32 (2021) no. 2, pp. 267-279.

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Let $M$ be an $H$-supplemented coatomic module with FIEP. Then we prove that $M$ is dual square free if and only if every maximal submodule of $M$ is fully invariant. Let $M=\bigoplus_{i\in I} M_i$ be a direct sum, such that $M$ is coatomic. Then we prove that $M$ is dual square free if and only if each $M_i$ is dual square free for all $i\in I$ and, $M_i$ and $\bigoplus_{j\neq i}M_j$ are dual orthogonal. Finally we study the endomorphism rings of dual square free modules. Let $M$ be a quasi-projective module. If $\operatorname{End}_R(M)$ is right dual square free, then $M$ is dual square free. In addition, if $M$ is finitely generated, then $\operatorname{End}_R(M)$ is right dual square free whenever $M$ is dual square free. We give several examples illustrating our hypotheses.
Keywords: dual square free module, endoregular module, (finite) internal exchange property.
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M. Medina-Bárcenas; D. Keskin Tütüncü; Y. Kuratomi. A study on dual square free modules. Algebra and discrete mathematics, Tome 32 (2021) no. 2, pp. 267-279. http://geodesic.mathdoc.fr/item/ADM_2021_32_2_a8/

[1] F. W. Anderson and K. R. Fuller, Rings and categories of modules, Springer-Verlag, New York, 1974 | MR | Zbl

[2] G. M. Brodskii and R. Wisbauer, “General distributivity and thickness of modules”, Arab. J. Sci. Eng., 25:2, Part C (2000), 95–128 | MR | Zbl

[3] J. Castro Pérez and J. Ríos Montes, “Prime submodules and local gabriel correspondence in $\sigma[M]$”, Comm. Algebra, 40 (2012), 213–232 | DOI | MR | Zbl

[4] J. Castro Pérez, M. Medina Bárcenas, J. Ríos Montes, and A. Zaldívar, “On semiprime Goldie modules”, Comm. Algebra, 44 (2016), 4749–4768 | DOI | MR | Zbl

[5] G. D'este and D. Keskin Tütüncü, “Pseudo projective modules which are not quasi-projective and quivers”, Taiwanese J. Math., 22 (2018), 1083–1090 | MR

[6] N. Ding, Y. Ibrahim, M. Yousif and Y. Zhou, “$D4$-modules”, J. Algebra Appl., 16:09 (2017), 1750166, 25 pp. | DOI | MR | Zbl

[7] A. Facchini, “Krull-Schmidt fails for serial modules”, Trans. Amer. Math. Soc., 348 (1996), 4561–4575 | DOI | MR | Zbl

[8] G. Güngöroğlu, “Coatomic modules”, Far East J. Math. Sciences, 2, Special Volume (1998), 153–162

[9] Y. Ibrahim and M. Yousif, “Rings whose injective hulls are dual square free”, Comm. Algebra, 48:3 (2020), 1011–1021 | DOI | MR | Zbl

[10] Y. Ibrahim and M. Yousif, “Dual-square-free modules”, Comm. Algebra, 47 (2019), 2954–2966 | DOI | MR | Zbl

[11] M. C. Izurdiaga, “Supplement submodules and a generalization of projective modules”, J. Algebra, 277 (2004), 689–702 | DOI | MR | Zbl

[12] D. Keskin Tütüncü, I. Kikumasa, Y. Kuratomi, and Y. Shibata, “On dual of square free modules”, Comm. Algebra, 46 (2018), 3365–3376 | DOI | MR | Zbl

[13] Y. Kuratomi, “Direct sums of $H$-supplemented modules”, J. Algebra Appl., 13 (2014) | DOI | MR | Zbl

[14] G. Lee, S. T. Rizvi and C. Roman, “Module whose endomorphism rings are von Neumann regular”, Comm. Algebra, 41 (2013), 4066–4088 | DOI | MR | Zbl

[15] M. Medina-Bárcenas and H. Sim, “Abelian endoregular modules”, J. Algebra Appl., 19:11 (2020) | DOI | MR

[16] K. M. Rangaswamy, and N. Vanaja, “Quasi-projectives in abelian and module categories”, Pacific J. Math., 43:1 (1972), 221–238 | DOI | MR | Zbl

[17] R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach, Reading, 1991 | MR | Zbl