Lattice of Subinjective Portfolios of Modules

Abstract

Given a ring $R$, we study its right subinjective profile $\mathfrak{siP}(R)$ to be the collection of subinjectivity domains of its right $R$-modules. We deal with the lattice structure of the class $\mathfrak{siP}(R)$. We show that the poset $(\mathfrak{siP}(R),\subseteq)$ forms a complete lattice, and an indigent $R$-module exists if $\mathfrak{siP}(R)$ is a set. In particular, if $R$ is a generalized uniserial ring with $J^{2}(R)=0$, then the lattice $(\mathfrak{siP}(R),\subseteq,\wedge, \vee)$ is Boolean.

Keywords

• Subinjectivity domain
• subinjective profile
• complete lattice of subinjectivity domains

References

1. B. Saraç, On rings whose quasi-injective modules are injective or semisimple, Journal of Algebra and Its Applications 21 (12) (2022) Article ID 2350005 23 pages.
2. Y. Durğun, An alternative perspective on flatness of modules, Journal of Algebra and Its Applications 15 (08) (2016) Article ID 1650145 18 pages.
3. J. Trlifaj, Whitehead test modules, Transactions of the American Mathematical Society 348 (4) (1996) 1521–1554.
4. S. Crivei, R. Pop, Projectivity and subprojectivity domains in exact categories, Journal of Algebra and Its Applications (2023) Article ID 2550134 19 pages.
5. C. Holston, S. R. Lopez-Permouth, N. O. Ertaş, Rings whose modules have maximal or minimal projectivity domain, Journal of Pure and Applied Algebra 216 (3) (2012) 480–494.
6. N. O. Ertaş, R. Tribak, Some variations of projectivity, Journal of Algebra and Its Applications 21 (12) (2022) Article ID 2250236 19 pages.
7. Y. Durğun, Subprojectivity domains of pure-projective modules, Journal of Algebra and Its Applications 19 (05) (2020) 2050091 14 pages.
8. A. Harmanci, S. R. Lopez-Permouth, B. Ungor, On the pure-injectivity profile of a ring, Communications in Algebra 43 (11) (2015) 4984–5002.

9. R. Alizade, Y. M. Demirci, B. N. Türkmen, E. Türkmen, On rings with one middle class of injectivity domains, Mathematical Communications 27 (1) (2022) 109–126.
10. S. E. Toksoy, Modules with minimal copure-injectivity domain, Journal of Algebra and Its Applications 18 (11) (2019) Article ID 1950201 14 pages.
11. S. R. Lopez-Permouth, J. Mastromatteo, Y. Tolooei, B. Ungor, Pure-injectivity from a different perspective, Glasgow Mathematical Journal 60 (1) (2018) 135–151.
12. P. Aydoğdu, S. R. Lopez-Permouth, An alternative perspective on injectivity of modules, Journal of Algebra 338 (1) (2011) 207–219.
13. R. Alizade, E. Büyükaşik, N. Er, Rings and modules characterized by opposites of injectivity, Journal of Algebra 409 (2014) 182–198.
14. T. Y. Lam, Lectures on modules and rings, Springer, New York, 1999.
15. R. Wisbauer, Foundations of module and ring theory, Gordon and Breach Science Publishers, Reading, 1991.
16. J. Rotman, An introduction to homological algebra, Academic Press, New York, 1979.
17. A. N. Alahmadi, M. Alkan, S. Lopez-Permouth, Poor modules: The opposite of injectivity, Glasgow Mathematical Journal 52 (A) (2010) 7–17.
18. F. Altinay, E. Büyükaşık, Y. Durğun, On the structure of modules defined by subinjectivity, Journal of Algebra and Its Applications 18 (10) (2019) 1950188 13 pages.
19. C. Holston, S. R. Lopez-Permouth, J. Mastromatteo, J. E. Simental-Rodriguez, An alternative perspective on projectivity of modules, Glasgow Mathematical Journal 57 (1) (2015) 83–99.
20. N. V. Dung, D. V. Huynh, P. F. Smith, R. Wisbauer, Extending modules, Longman Scientific and Technical, Harlow, 1994.
21. G. Calugareanu, Lattice concepts of module theory, Springer Science and Business Media, Dordrecht, 2013.