Lattice of Subinjective Portfolios of Modules

Year 2024, Issue: 47, 11 - 19, 30.06.2024

Yilmaz Durğun

https://doi.org/10.53570/jnt.1467235

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

Ethical Statement

The author declares no conflict of interest.

Supporting Institution

The Scientific and Technological Research Council of T\"{u}rkiye (TUBITAK)

Project Number

122F130

References

• 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.
• Y. Durğun, An alternative perspective on flatness of modules, Journal of Algebra and Its Applications 15 (08) (2016) Article ID 1650145 18 pages.
• J. Trlifaj, Whitehead test modules, Transactions of the American Mathematical Society 348 (4) (1996) 1521–1554.
• S. Crivei, R. Pop, Projectivity and subprojectivity domains in exact categories, Journal of Algebra and Its Applications (2023) Article ID 2550134 19 pages.
• 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.
• N. O. Ertaş, R. Tribak, Some variations of projectivity, Journal of Algebra and Its Applications 21 (12) (2022) Article ID 2250236 19 pages.
• Y. Durğun, Subprojectivity domains of pure-projective modules, Journal of Algebra and Its Applications 19 (05) (2020) 2050091 14 pages.
• A. Harmanci, S. R. Lopez-Permouth, B. Ungor, On the pure-injectivity profile of a ring, Communications in Algebra 43 (11) (2015) 4984–5002.
• 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.
• S. E. Toksoy, Modules with minimal copure-injectivity domain, Journal of Algebra and Its Applications 18 (11) (2019) Article ID 1950201 14 pages.
• S. R. Lopez-Permouth, J. Mastromatteo, Y. Tolooei, B. Ungor, Pure-injectivity from a different perspective, Glasgow Mathematical Journal 60 (1) (2018) 135–151.
• P. Aydoğdu, S. R. Lopez-Permouth, An alternative perspective on injectivity of modules, Journal of Algebra 338 (1) (2011) 207–219.
• R. Alizade, E. Büyükaşik, N. Er, Rings and modules characterized by opposites of injectivity, Journal of Algebra 409 (2014) 182–198.
• T. Y. Lam, Lectures on modules and rings, Springer, New York, 1999.
• R. Wisbauer, Foundations of module and ring theory, Gordon and Breach Science Publishers, Reading, 1991.
• J. Rotman, An introduction to homological algebra, Academic Press, New York, 1979.
• A. N. Alahmadi, M. Alkan, S. Lopez-Permouth, Poor modules: The opposite of injectivity, Glasgow Mathematical Journal 52 (A) (2010) 7–17.
• 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.
• 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.
• N. V. Dung, D. V. Huynh, P. F. Smith, R. Wisbauer, Extending modules, Longman Scientific and Technical, Harlow, 1994.
• G. Calugareanu, Lattice concepts of module theory, Springer Science and Business Media, Dordrecht, 2013.