Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-24T03:51:28.507Z Has data issue: false hasContentIssue false
  • English
  • Français

On Modules of Singular Submodule Zero

Published online by Cambridge University Press:  20 November 2018

Vasily C. Cateforis  and
Francis L. Sandomierski
Vasily C. Cateforis
Affiliation:
University of Kentucky, Lexington, Kentucky
Francis L. Sandomierski
Affiliation:
Kent State University, Kent, Ohio
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we generalize to modules of singular submodule zero over a ring of singular ideal zero some of the results, which are well known for torsion-free modules over a commutative integral domain, e.g. [2, Chapter VII, p. 127], or over a ring, which possesses a classical right quotient ring, e.g. [13, § 5].

Let R be an associative ring with 1 and let M be a unitary right R-module, the latter fact denoted by MR. A submodule NR of MR is large in MR (MR is an essential extension of NR) if NR intersects non-trivially every non-zero submodule of MR; the notation NR ⊆′ MR is used for the statement “NR is large in MR” The singular submodule of MR, denoted Z(MR), is then defined to be the set {m ∈ M| r(m) ⊆’ RR}, where

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 2 , April 1971 , pp. 345 - 354
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Bass, H., Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc. 95 (1960), 466–488.Google Scholar
2. Cartan, H. and Eilenberg, S., Homological algebra (Princeton Univ. Press, Princeton, N.J., 1956).Google Scholar
3. Cateforis, V., On regular self-injective rings, Pacific J. Math. 30 (1969), 39–45.Google Scholar
4. Cateforis, V., Flat regular quotient rings, Trans. Amer. Math. Soc. 188 (1969), 241–249.Google Scholar
5. Cateforis, V., Two-sided semisimple maximal quotient rings, Trans. Amer. Math. Soc. 149 (1970), 339–350.Google Scholar
6. Chase, S. U., Direct product of modules, Trans. Amer. Math. Soc. 97 (1960), 457–473.Google Scholar
7. Colby, R. and Rutter, E., Semi-primary QF-3 rings, Nagoya Math. J. 32 (1968), 253–257.Google Scholar
8. Fuller, K., The structure of QF-3 rings, Trans. Amer. Math. Soc. 134 (1968), 343–354.Google Scholar
9. Harada, M., QF-3 and semi-primary PP-rings. I, Osaka J. Math. 2 (1965), 357–368.Google Scholar
10. Harada, M., QF-3 and semi-primary PP-rings. II, Osaka J. Math. 3 (1966), 21–27.Google Scholar
11. Harada, M., Hereditary semi-primary rings and triangular matrix rings, Nagoya Math. J. 27 (1966), 463–484.Google Scholar
12. Lambek, J., Lectures on rings and modules (Blaisdell, Waltham, Massachusetts, 1966).Google Scholar
13. Levy, L., Torsion-free and divisible modules over non-integral domains, Can. J. Math. 15 (1963), 132–151.Google Scholar
14. Osofsky, B., Cyclic infective modules of full linear rings, Proc. Amer. Math. Soc. 17 (1966), 247–253.Google Scholar
15. Sandomierski, F. L., Semi-simple maximal quotient rings, Trans. Amer. Math. Soc. 128 (1967), 112–120.Google Scholar
16. Sandomierski, F. L., Nonsingular rings, Proc. Amer. Math. Soc. 19 (1968), 225–230.Google Scholar
17. Wei, D. Y., On the concept of torsion and divisibility for general rings, Illinois J. Math. 13 (1969), 414–431.Google Scholar