[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.
[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.
Rings for which every cosingular module is discrete
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.
[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.
[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.
Talebi, Y., Moniri Hamzekolaee, A. R., Harmancı, A., Üngör, B. (2020). Rings for which every cosingular module is discrete. Hacettepe Journal of Mathematics and Statistics, 49(5), 1635-1648. https://doi.org/10.15672/hujms.500759
AMA
Talebi Y, Moniri Hamzekolaee AR, Harmancı A, Üngör B. Rings for which every cosingular module is discrete. Hacettepe Journal of Mathematics and Statistics. October 2020;49(5):1635-1648. doi:10.15672/hujms.500759
Chicago
Talebi, Yahya, Ali Reza Moniri Hamzekolaee, Abdullah Harmancı, and Burcu Üngör. “Rings for Which Every Cosingular Module Is Discrete”. Hacettepe Journal of Mathematics and Statistics 49, no. 5 (October 2020): 1635-48. https://doi.org/10.15672/hujms.500759.
EndNote
Talebi Y, Moniri Hamzekolaee AR, Harmancı A, Üngör B (October 1, 2020) Rings for which every cosingular module is discrete. Hacettepe Journal of Mathematics and Statistics 49 5 1635–1648.
IEEE
Y. Talebi, A. R. Moniri Hamzekolaee, A. Harmancı, and B. Üngör, “Rings for which every cosingular module is discrete”, Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 5, pp. 1635–1648, 2020, doi: 10.15672/hujms.500759.
ISNAD
Talebi, Yahya et al. “Rings for Which Every Cosingular Module Is Discrete”. Hacettepe Journal of Mathematics and Statistics 49/5 (October 2020), 1635-1648. https://doi.org/10.15672/hujms.500759.
JAMA
Talebi Y, Moniri Hamzekolaee AR, Harmancı A, Üngör B. Rings for which every cosingular module is discrete. Hacettepe Journal of Mathematics and Statistics. 2020;49:1635–1648.
MLA
Talebi, Yahya et al. “Rings for Which Every Cosingular Module Is Discrete”. Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 5, 2020, pp. 1635-48, doi:10.15672/hujms.500759.
Vancouver
Talebi Y, Moniri Hamzekolaee AR, Harmancı A, Üngör B. Rings for which every cosingular module is discrete. Hacettepe Journal of Mathematics and Statistics. 2020;49(5):1635-48.