Weakly Poor Modules

Year 2022, Volume: 10 Issue: 2, 250 - 254, 31.10.2022

Yusuf Alagöz

Abstract

In this paper, weakly poor modules are introduced as modules whose injectivity domains are contained in the class of all copure-split modules. This notion gives a generalization of both poor modules and copure-injectively poor modules. Properties involving weakly poor modules as well as examples that show the relations between weakly poor modules, poor modules, impecunious modules and copure-injectively poor modules are given. Rings over which every module is weakly poor are right CDS. A ring over which there is a cyclic projective weakly poor module is proved to be weakly poor. Moreover, the characterizations of weakly poor abelian groups is given. It states that an abelian group $A$ is weakly poor if and only if $A$ is impecunious if and only if for every prime integer $p$, $A$ has a direct summand isomorphic to $\mathbb{Z}_{p^{n}}$ for some positive integer $n$. Consequently, an example of a weakly poor abelian group which is neither poor nor copure-injectively poor is given so that the generalization defined is proper.

Keywords

Copure-injective modules, copure-split modules, (weakly) poor modules, CDS rings

References

• [1] Alag¨oz, Y., Relative subcopure-injective modules, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 69(1), 832-846 (2020).
• [2] Anderson, F.W., Fuller, K. R., Rings and categories of modules, Springer-Verlag, New York, 1974.
• [3] Alahmadi, A. N., Alkan, M., L´opez-Permouth, S. R., Poor modules: The opposite of injectivity, Glasg. Math. J., 52(A), 7-17 (2010).
• [4] Alizade, R., B¨uy¨ukas¸ık, E., Poor and pi-poor abelian groups, Comm. Algebra, 45(1), 420-427 (2017).
• [5] Demirci, Y. M., Modules and abelian groups with a bounded domain of injectivity, J. Algebra Appl., 16(2), 1850108 (2018).
• [6] Demirci, Y. M., Nis¸ancı T¨urkmen, B., T¨urkmen, E., Rings with modules having a restricted injectivity domain, S˜ao Paulo J. Math. Sci. 14, 312-326 (2020).
• [7] Fieldhouse, D. J., Pure theories, Math. Ann., 184, 1-18 (1969).
• [8] Harmanci, A., Lo´pez-Permouth, S. R., U¨ ngo¨r, B., On the pure-injectivity profile of a ring, Comm. Algebra, 43(11), 4984-5002 (2015).
• [9] Hiremath, V. A., Cofinitely generated and cofinitely related modules, Acta Math. Acad. Sci. Hungar., 39, 1-9 (1982).
• [10] Hiremath (Madurai), V. A., Copure Submodules, Acta Math. Hung., 44(1-2), 3-12 (1984).
• [11] Hiremath (Madurai), V. A., Copure-injective modules, Indian J. Pure Appl. Math., 20(3), 250-259 (1989).
• [12] Jans, J. P., On co-noetherian rings, J. London Math. Soc., 1, 588-590 (1969).
• [13] Maurya, S. K., Toksoy, S. E., Copure-direct-injective modules, J. Algebra Appl., 21(9), 2250187 (2022).
• [14] Mohamed, S. H., M¨uller, B. J., Continuous and discrete modules, London Mathematical Society Lecture Note 147 (Cambridge University Press), Cambridge 1990.
• [15] Sharpe, D.W., Vamos, P., Injective Modules, Cambridge Tracts in Mathematics and Mathematical Physics, 62, Cambridge. 1972.
• [16] Toksoy, S. E., Modules with minimal copure-injectivity domain, J. Algebra Appl., 18(11), 1950201 (2019).
• [17] Vamos, P., The dual of the notion of “finitely generated”, J. London Math. Soc., 43, 643-646 (1968).
• [18] Vamos, P., Classical rings, J. Algebra, 34, 114-129 (1975).