Relative subcopure-injective modules

Abstract

In this paper, copure-injective modules is examined from an alternative perspective. For two modules A and B, A is called B-subcopure-injective if for every copure monomorphism f : B → C and homomorphism g : B → A, there exists a homomorphism h : C → A  such that hf=g. For a module A, the subcopure-injectivity domain of A is defined to be the collection of all modules B such that A is B-subcopure-injective. Basic properties of the notion of subcopure-injectivity are investigated. We obtain characterizations for various types of rings and modules, including copure-injective modules, right CDS rings and right V-rings in terms of subcopure-injectivity domains. Since subcopure-injectivity domains clearly contains all copure-injective modules, studying the notion of modules which are subcopure-injective only with respect to the class of copure-injective modules is reasonable. We refer to these modules as sc-indigent. We studied the properties of subcopure-injectivity domains and of sc-indigent modules and investigated over some certain rings.

Keywords

• Copure-injective modules,subcopure-injectivity domains,sc-indigent modules,CDS rings

References

1. Anderson, F. W., Fuller, K. R., Rings and categories of modules, Springer-Verlag, New York, 1974.
2. Aydoğdu, P., López-Permouth, S. R., An alternative perspective on injectivity of modules, J. Algebra, 338 (2011) 207-219.
3. Durğun, Y., An alternative perspective on flatness of modules, J. Algebra Appl., 15(8) (2016) 1650145, 18. communiserial : Fieldhouse, D. J., Pure theories, Math. Ann., 184 (1969) 1-18.
4. Harmanci, A., López-Permouth, S. R., Üngör, B., On the pure-injectivity profile of a ring, Comm. Algebra , 43(11) (2015) 4984-5002.
5. Hiremath, V. A., Cofinitely generated and cofinitely related modules, Acta Math. Acad. Sci. Hungar., 39 (1982), 1-9.
6. Hiremath (Madurai), V. A., Copure Submodules, Acta Math. Hung., 44(1-2) (1984) 3-12.
7. Hiremath (Madurai), V. A., Copure-injective modules, Indian J. Pure Appl. Math., 20(3) (1989) 250-259.
8. Hiremath (Madurai), V. A., Co-absolutely co-pure modules, Proceedings of the Edinburgh Mathematical Society, 29 (1986), 289-298.

9. Jans, J. P., On co-noetherian rings, J. London Math. Soc., 1 (1969), 588-590.
10. López-Permouth, S. R., Mastromatteo, J., Tolooei, Y., Üngör, B., Pure-injectivity from a different perspective, Glasg. Math. J., 60(1) (2018), 135--151.
11. López-Permouth, S. R., Simental-Rodriguez, J. E., Characterizing rings in terms of the extent of the injectivity and projectivity of their modules, J. Algebra, 362 (2012), 56-69.
12. Mao, L., Ding, N., Notes On Cotorsion Modules, Comm. Algebra, 33 (2005), 349-360.
13. Sharpe, D. W., Vamos, P., Injective Modules, (Cambridge Tracts in Mathematics and Mathematical Physics, 62), Cambridge, 1972.
14. Toksoy, S. E., Modules with minimal copure-injectivity domain, J. Algebra Appl., 18(11) (2019), 195-201.
15. Vamos, P., The dual of the notion of finitely generated, J. London Math. Soc., 43 (1968), 643-646.
16. Vamos, P., Classical rings, J. Algebra, 34 (1975), 114-129.