On Rings Whose Quasi-Projective Modules Are Projective or Semisimple
DOI:
https://doi.org/10.26713/cma.v12i2.1490Keywords:
Projective module, p-poor module, Projectivity domain, Semi-Artininan ringAbstract
For two modules \(M\) and \(N\), \(P_M(N)\) stands for the largest submodule of \(N\) relative to which \(M\) is projective. For any module \(M\), \(P_M(N)\) defines a left exact preradical. It is given some properties of \(P_M(N)\).\ We express \(P_M(N)\) as a trace submodule. In this paper, we study rings with no quasi-projective modules other than semisimples and projectives, that is, rings whose quasi-projectives are either projective or semisimple (namely QPS-ring). Semi-Artinian rings and rings with no right p-middle class are characterized by using this functor: a ring \(R\) right semi-Artinian if and only if for any right \(R\)-module \(M\), \(P_M(M)\leq_e M\).Downloads
References
A.N. Alahmadi, M. Alkan and S.R. López-Permouth, Poor modules: the opposite of injectivity, Glasgow Mathematical Journal 52(A) (2010), 7 – 17, DOI: 10.1017/S001708951000025X.
F.W. Anderson and K.R. Fuller, Rings and Categories of Modules, Springer-Verlag, New York (1974).
P. Aydogdu and B. Saraç, On Artinian rings with restricted class of injectivity domains, Journal of Algebra 377 (2013), 49 – 65, DOI: 10.1016/j.jalgebra.2012.11.027.
H.Q. Dinh, C.J. Holston and D.V. Huynh, Quasi-projective modules over prime hereditary noetherian V-rings are projective or injective, Journal of Algebra 360 (2012), 87 – 91, DOI: 10.1016/j.jalgebra.2012.04.002.
N. Er, S.R. López-Permouth and N. Sökmez, Rings whose modules have maximal or minimal injectivity domains, Journal of Algebra 330(1) (2011), 404 – 417, DOI: 10.1016/j.jalgebra.2010.10.038.
N. Er, Rings characterized via a class of left exact preradicals, Proceedings of the Edinburgh Mathematical Society 59(3) (2016), 641 – 653, DOI: 10.1017/S0013091515000206.
A. Facchini, Module Theory, Endomorphism rings and direct sum decompositions in some classes of modules, Birkhäuser-Verlag (1998).
K. Goodearl, Singular torsion and the splitting properties, Mem. Amer. Math. Soc. 124 (1972).
C. Holston, S.R. Lopez-Permouth and N.O. Erta¸s, Rings whose modules have maximal or minimal projectivity domain, Journal of Pure and Applied Algebra 216 (2012), 673 – 678, DOI: 10.1016/j.jpaa.2011.08.002.
S.R. López-Permouth and J.E. Simental, Characterizing rings in terms of the extent of the injectivity and projectivity of their modules, Journal of Algebra 362 (2012), 56 – 69, DOI: 10.1016/j.jalgebra.2012.04.005.
S.H. Mohamed and B.J. Müller, Continuous and discrete modules, London Mathematical Society Lecture Note Series, 147, Cambridge University Press, Cambridge (1990).
K.M. Rangaswamy and N. Vanaja, Quasi projectives in abelian and module categories, Pacific Journal of Mathematics 43(1) (1972), 221 – 238, https://projecteuclid.org/download/pdf1/euclid.pjm/1102959656.
B. Saraç, On rings whose quasi-injective modules are injective or semisimple, https://archive.org/details/qis-rings.
Downloads
Published
How to Cite
Issue
Section
License
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a CCAL that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.
