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q-Hypercyclic Rings

Published online by Cambridge University Press:  20 November 2018

S. K. Jain  and
D. S. Malik
S. K. Jain
Affiliation:
Ohio University, Athens, Ohio
D. S. Malik
Affiliation:
Ohio University, Athens, Ohio
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A ring R is called q-hypercyclic (hypercyclic) if each cyclic ring R-module has a cyclic quasi-injective (injective) hull. A ring R is called a qc-ring if each cyclic right R-module is quasi-injective. Hypercyclic rings have been studied by Caldwell [4], and by Osofsky [12]. A characterization of qc-rings has been given by Koehler [10]. The object of this paper is to study q-hypercyclic rings. For a commutative ring R, R can be shown to be q-hypercyclic (= qc-ring) if R is hypercyclic. (Theorems 4.2 and 4.3). Whether a hypercyclic ring (not necessarily commutative) is q-hypercyclic is considered in Theorem 3.11 by showing that a local hypercyclic ring R is q-hypercyclic if and only if the Jacobson radical of R is nil. However, we do not know if there exists a local hypercyclic ring with nonnil radical [12]. Example 3.10 shows that a q-hypercyclic ring need not be hypercyclic.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 37 , Issue 3 , 01 June 1985 , pp. 452 - 466
Copyright
Copyright © Canadian Mathematical Society 1985

References

1. Ahsan, J., Rings all whose cyclic modules are quasi-injectives, Proc. London Math. Soc. 27 (1973), 425–439.Google Scholar
2. Anderson, F. W. and Fuller, K. R., Rings and categories of modules (Springer-Verlag, 1974).CrossRefGoogle Scholar
3. Bass, H., Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc. (1960), 466–488.Google Scholar
4. Caldwell, W., Hypercyclic rings, Pacific J. Math. 24 (1968), 29–44.Google Scholar
5. Faith, C., Algebra II (Springer-Verlag, 1976).Google Scholar
6. Faith, C., On Köthe rings, Math. Ann. 164 (1966), 207–212.Google Scholar
7. Goel, V. K. and Jain, S. K., π-injective modules and rings whose cyclics are π-injectives, Comm. Alg. 6 (1978), 59–73.Google Scholar
8. Ikeda, M. and Nakayama, T., On some characteristic properties of quasi-Frobenious and regular rings, Proc. Amer. Math. Soc. 5 (1954), 15–19.Google Scholar
9. Klatt, G. B. and Levy, L. S., Pre-selfinjective rings, Trans. Amer. Math. Soc. 137 (1969), 407–419.Google Scholar
10. Koehler, A., Rings with quasi-injective cyclic modules, Quart. J. Math. 25 (1974), 51–55.Google Scholar
11. Koehler, A., Quasi-projective covers and direct sums, Proc. Amer. Math. Soc. 24 (1970), 655–658.Google Scholar
12. Osofsky, B. L., Noncommutative rings whose cyclic modules have cyclic injective hull, Pacific J. Math. 25 (1968), 331–340.Google Scholar
13. Sharpe, D. W. and Vamos, P., Injective modules (Cambridge, 1972).Google Scholar