Geodesic
AN ALTERNATIVE PERSPECTIVE ON PROJECTIVITY OF MODULES
Glasgow mathematical journal, Tome 57 (2015) no. 1, pp. 83-99

Voir la notice de l'article provenant de la source Cambridge University Press

We approach the analysis of the extent of the projectivity of modules from a fresh perspective as we introduce the notion of relative subprojectivity. A module M is said to be N-subprojective if for every epimorphism g : B → N and homomorphism f : M → N, there exists a homomorphism h : M → B such that gh = f. For a module M, the subprojectivity domain of M is defined to be the collection of all modules N such that M is N-subprojective. We consider, for every ring R, the subprojective profile of R, namely, the class of all subprojectivity domains for R modules. We show that the subprojective profile of R is a semi-lattice, and consider when this structure has coatoms or a smallest element. Modules whose subprojectivity domain is as smallest as possible will be called subprojectively poor (sp-poor) or projectively indigent (p-indigent), and those with co-atomic subprojectivy domain are said to be maximally subprojective. While we do not know if sp-poor modules and maximally subprojective modules exist over every ring, their existence is determined for various families. For example, we determine that artinian serial rings have sp-poor modules and attain the existence of maximally subprojective modules over the integers and for arbitrary V-rings. This work is a natural continuation to recent papers that have embraced the systematic study of the injective, projective and subinjective profiles of rings.
DOI : 10.1017/S0017089514000135
Mots-clés : 16D40, 16D50, 16D80 
HOLSTON, CHRIS; LÓPEZ-PERMOUTH, SERGIO R.; MASTROMATTEO, JOSEPH; SIMENTAL-RODRÍGUEZ, JOSÉ E. AN ALTERNATIVE PERSPECTIVE ON PROJECTIVITY OF MODULES. Glasgow mathematical journal, Tome 57 (2015) no. 1, pp. 83-99. doi: 10.1017/S0017089514000135
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[1] 1.Alahmadi, A. N., Alkan, M. and López-Permouth, S. R., Poor modules, the opposite of injectivity, Glasgow Math. J. 52A (2010), 7–17. Google Scholar | DOI

[2] 2.Amin, I., Yousif, M. and Zeyada, N., Soc-injective rings and modules, Comm. Algebra 33 (11) (2005), 4229–4250. Google Scholar | DOI

[3] 3.Anderson, F. W. and Fuller, K. R., Rings and categories of modules (Springer-Verlag, New York, NY, 1992). Google Scholar | DOI

[4] 4.Assem, I., Simson, D. and Skowroński, A., Elements of the representation theory of associative algebras. Vol. 1: Techniques of representation theory, London Math. Soc. Student Texts, vol. 65 (Cambridge University Press, Cambridge, 2006). Google Scholar | DOI

[5] 5.Aydoğdu, P. and López-Permouth, S. R., An alternative perspective on injectivity of modules, J. Algebra 338(2011), 207–219. Google Scholar | DOI

[6] 6.Baer, R., Abelian groups without elements of finite order, Duke Math. J. 3 (1) (1937), 68–122. Google Scholar | DOI

[7] 7.Bass, H., Finitistic dimension and a homological characterization of semiprimary rings, Trans. Amer. Math. Soc. 95 (1960), 466–488. Google Scholar | DOI

[8] 8.Chase, S. U., Direct products of modules, Trans. Amer. Math. Soc. 97 (3) (1960), 457–473. Google Scholar | DOI

[9] 9.Clark, J., Lomp, C., Vanaja, N. and Wisbauer, R., Lifting modules: Supplements and projectivity in module theory, Frontiers in Mathematics (Birkhäuser-Verlag, Basel, Switzerland, 2006). Google Scholar

[10] 10.Er, N., López-Permouth, S. R. and Sökmez, N., Rings whose modules have maximal or minimal injectivity domains, J. Algebra 330 (1) (2011), 404–417. Google Scholar

[11] 11.Faith, C., Algebra II: Ring theory, Grundlehren Math. Wiss., vol. 191 (Springer-Verlag, New York, NY, 1976). Google Scholar | DOI

[12] 12.Fuchs, L., Infinite Abelian groups, vol. 2 (Academic Press, Waltham, MA, 1973). Google Scholar

[13] 13.Holston, C., López-Permouth, S. R. and Orhan Ertaş, N., Rings whose modules have maximal or minimal projectivity domains, J. Pure Appl. Algebra 216 (3) (2012), 673–678. Google Scholar

[14] 14.Lam, T. Y., Lectures on modules and rings (Springer, Berlin, Germany, 1999). Google Scholar | DOI

[15] 15.López-Permouth, S. R. and Simental, J. E., Characterizing rings in terms of the extent of the injectivity and projectivity of their modules, J. Algebra 362 (2012), 56–69. Google Scholar | DOI

[16] 16.Mohamed, S. H. and Müller, B. J., Continuous and discrete modules (Cambridge University Press, Cambridge, UK, 1990). Google Scholar

[17] 17.Stenström, B., Rings of quotients, (Springer-Verlag, Berlin, Germany, 1975). Google Scholar

[18] 18.Teply, M., Right hereditary, right perfect rings are semiprimary, Advances in Ring Theory, Trends in Mathematics (Birkhäuser, Boston, MA, 1996), 313–316. Google Scholar

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