Coretractable modules relative to a submodule
Keywords:
Coretractable module, N-coretractable module
Abstract
Let $R$ be a ring and $M$ a right $R$-module. Let $N$ be a proper submodule of $M$. We say that $M$ is $N$-coretractable (or $M$ is coretractable relative to $N$) provided that, for every proper submodule $K$ of $M$ containing $N$, there is a nonzero homomorphism $f:M/K\rightarrow M$. We present some conditions that a module $M$ is coretractable if and only if $M$ is coretractable relative to a submodule $N$. We also provide some examples to illustrate special cases.
References
A. N. Abyzov, A. A. Tuganbaev, Retractable and coretractable modules, J. Math. Sci. 213(2) (2016) 132–142.
B. Amini, M. Ershad, H. Sharif, Coretractable modules, J. Aust. Math. Soc. 86(3) (2009) 289–304.
F. W. Anderson, K. R. Fuller, Rings and Categories of Modules, Springer-Verlog, New York, 1992.
N. O. Ertas, D. K. Tütüncü, R. Tribak, A variation of coretractable modules, Bull. Malays. Math. Sci. Soc. 41(3) (2018) 1275–1291.
S. M. Khuri, Endomorphism rings and lattice isomorphisms, J. Algebra 56(2) (1979) 401–408.
S. M. Khuri, Nonsingular retractable modules and their endomorphism rings, Bull. Aust. Math. Soc. 43(1) (1991) 63–71.
T. Y. Lam, Lectures on Modules and Rings, Springer-Verlag, New York, 1999.
G. Lee, S. T. Rizvi, C. S. Roman, Dual Rickart modules, Comm. Algebra 39(11) (2011) 4036-4058.
S. H. Mohamed, B. J. Müller, Continuous and Discrete Modules, London Math. Soc. Lecture Notes Series 147, Cambridge, University Press, Cambridge, 1990.
A. R. M. Hamzekolaee, A generalization of coretractable modules, J. Algebraic Syst. 5(2) (2017) 163–176.
R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach, Philadelphia, 1991.
J. M. Zelmanowitz, Correspondences of closed submodules, Proc. Amer. Math. Soc. 124(10) (1996) 2955–2960.
J. Žemlicka, Completely coretractable rings, Bull. Iranian Math. 39(3) (2013) 523–528.
Z. Zhengping, A lattice isomorphism theorem for nonsingular retractable modules, Canad. Math. Bull. 37(1) (1994) 140–144.
Y. Zhou, Generalizations of perfect, semiperfect, and semiregular rings, Algebra Colloq. 7(3) (2000) 305–318.
B. Amini, M. Ershad, H. Sharif, Coretractable modules, J. Aust. Math. Soc. 86(3) (2009) 289–304.
F. W. Anderson, K. R. Fuller, Rings and Categories of Modules, Springer-Verlog, New York, 1992.
N. O. Ertas, D. K. Tütüncü, R. Tribak, A variation of coretractable modules, Bull. Malays. Math. Sci. Soc. 41(3) (2018) 1275–1291.
S. M. Khuri, Endomorphism rings and lattice isomorphisms, J. Algebra 56(2) (1979) 401–408.
S. M. Khuri, Nonsingular retractable modules and their endomorphism rings, Bull. Aust. Math. Soc. 43(1) (1991) 63–71.
T. Y. Lam, Lectures on Modules and Rings, Springer-Verlag, New York, 1999.
G. Lee, S. T. Rizvi, C. S. Roman, Dual Rickart modules, Comm. Algebra 39(11) (2011) 4036-4058.
S. H. Mohamed, B. J. Müller, Continuous and Discrete Modules, London Math. Soc. Lecture Notes Series 147, Cambridge, University Press, Cambridge, 1990.
A. R. M. Hamzekolaee, A generalization of coretractable modules, J. Algebraic Syst. 5(2) (2017) 163–176.
R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach, Philadelphia, 1991.
J. M. Zelmanowitz, Correspondences of closed submodules, Proc. Amer. Math. Soc. 124(10) (1996) 2955–2960.
J. Žemlicka, Completely coretractable rings, Bull. Iranian Math. 39(3) (2013) 523–528.
Z. Zhengping, A lattice isomorphism theorem for nonsingular retractable modules, Canad. Math. Bull. 37(1) (1994) 140–144.
Y. Zhou, Generalizations of perfect, semiperfect, and semiregular rings, Algebra Colloq. 7(3) (2000) 305–318.
