On Rings Whose Quasi-Projective Modules Are Projective or Semisimple

Nil Orhan Ertas, Ummahan Acar

Abstract


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 \(M\), \(P_M(M)\leq_e M\).

Keywords


Projective module; p-poor module; Projectivity domain; Semi-Artininan ring

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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.




DOI: http://dx.doi.org/10.26713%2Fcma.v12i2.1490

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