Rings for which every cosingular module is discrete

Year 2020, Volume: 49 Issue: 5, 1635 - 1648, 06.10.2020

Yahya Talebi , Ali Reza Moniri Hamzekolaee , Abdullah Harmancı , Burcu Üngör

https://doi.org/10.15672/hujms.500759

Abstract

In this paper we introduce the concepts of $CD$-rings and $CD$-modules. Let $R$ be a ring and $M$ be an $R$-module. We call $R$ a $CD$-ring in case every cosingular $R$-module is discrete, and $M$ a $CD$-module if every $M$-cosingular $R$-module in $\sigma[M]$ is discrete. If $R$ is a ring such that the class of cosingular $R$-modules is closed under factor modules, then it is proved that $R$ is a $CD$-ring if and only if every cosingular $R$-module is semisimple. The relations of $CD$-rings are investigated with $V$-rings, $GV$-rings, $SC$-rings, and rings with all cosingular $R$-modules projective. If $R$ is a semilocal ring, then it is shown that $R$ is right $CD$ if and only if $R$ is left $SC$ with $Soc(_{R}R)$ essential in $_{R}R$. Also, being a $V$-ring and being a $CD$-ring coincide for local rings. Besides of these, we characterize $CD$-modules with finite hollow dimension.

Keywords

CD-module , CD-ring , cosingular module , discrete module , V -ring , semilocal module , finite hollow dimension

References

• [1] F.W. Anderson and K.R. Fuller, Rings and Categories of Modules, Springer-Verlag, New York, 1992.
• [2] U.S. Chase, Direct product of modules, Trans. Amer. Math. Soc. 97, 457-473, 1960.
• [3] J. Clark, C. Lomp, N. Vanaja, and R. Wisbauer, Lifting Modules, Supplements and Projectivity in Module Theory, Frontiers in Math., Boston, Birkhäuser, 2006.
• [4] K.R. Goodearl, Singular Torsion and the Splitting Properties, Mem. Amer. Math. Soc., No. 124, 1972.
• [5] M.A. Kamal and A. Yousef, On principally lifting modules, Int. Electron. J. Algebra 2, 127-137, 2007.
• [6] D. Keskin and R. Tribak, When M-cosingular modules are projective, Vietnam J. Math. 33 (2), 214–221, 2005.
• [7] C. Lomp, On semilocal modules and rings, Comm. Algebra 27 (4), 1921-1935, 1999.
• [8] S.H. Mohamed and B.J. Müller, Continuous and Discrete Modules, in: London Math. Soc. Lecture Notes Series 147, Cambridge, University Press, 1990.
• [9] S. Mohamed and S. Singh, Generalizations of decomposition theorems known over perfect rings, J. Austral. Math. Soc. Ser. A 24, 496–510, 1977.
• [10] A.C. Ozcan, The torsion theory cogenerated by $\delta$-$M$-small modules and GCOmodules, Comm. Algebra 35 (2), 623–633, 2007.
• [11] S.T. Rizvi and M.F. Yousif, On continuous and singular modules, in: Non- Commutative Ring Theory, Lecture Notes in Mathematics Vol. 1448, 116-124, Springer, Berlin, Heidelberg, 1990.
• [12] N.V. Sanh, On SC-modules, Bull. Aust. Math. Soc. 48, 251-255, 1993.
• [13] B. Sarath and K. Varadarajan, Dual Goldie dimension - II, Comm. Algebra 7 (17), 1885-1899, 1979.
• [14] Y. Talebi, A.R.M. Hamzekolaee, M. Hosseinpour, A. Harmanci, and B. Ungor, Rings for which every cosingular module is projective, Hacet. J. Math. Stat. 48 (4), 973-984, 2019.
• [15] Y. Talebi and N. Vanaja, The torsion theory cogenerated by M-small modules, Comm. Algebra 30 (3), 1449-1460, 2002.
• [16] R. Tribak and D. Keskin, On $\overline{Z}_M$-semiperfect modules, East-West J. Math. 8 (2), 193-203, 2006.
• [17] B. Ungor, S. Halicioglu, and A. Harmanci, On a class of ⊕-supplemented modules, in: Ring Theory and Its Applications, Contemp. Math. 609, 123–136, Amer. Math. Soc., Providence, RI, 2014.
• [18] R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach, Reading, 1991.
• [19] H. Zöschinger, Koatomare moduln, Math. Z. 170, 221-232, 1980.