Abstract
The aim of this paper is to generalize the notion of pseudo-valuation to modules over arbitrary commutative rings. We generalize the notion of strongly prime ideal, as defined in Badawi et al. (Lecture Notes in Pure and Applied Mathematics 185:57–67, 1997, to the notion of strongly prime submodule. We define a module M to be a pseudo-valuation module if every prime submodule of M is strongly prime. It is shown that if M has a maximal submodule N, then M is pseudo-valuation if and only if N is strongly prime. Also, we characterize strongly prime submodules in pseudo-valuation modules. We investigate some properties of these modules, and study relations between some structures and these modules.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Anderson, D.F., Badawi, A., Dobbs, D.E.: Pseudo-valuation rings II. Boll. Unione Mat. Ital. 3-B, 535–545 (2000)
Badawi, A., Anderson, D.F., Dobbs, D.E.: Pseudo-valuation rings. In: Cahen, P.-J., et al. (eds.) Commutative Ring Theory. Lecture Notes Pure Appl. Math., vol. 185, pp. 57–67. Marcel Dekker, New York/Basel (1997)
Badawi, A.: On pseudo-almost valuation domains. Commun. Algebra 35, 1167–1181 (2007)
Badawi, A.: Remarks on pseudo-valuation rings. Commun. Algebra 28, 2343–2358 (2000)
Badawi, A.: A visit to valuation and pseudo-valuation domains. In: Anderson, D.E., Dobbs, D. (eds.) Zero-Dimensional Commutative Rings. Lecture Notes Pure Appl. Math., vol. 171, pp. 155–161. Marcel Dekker, New York/Basel (1995)
Badawi, A., Houston, E.: Powerful ideals, strongly primary ideals, almost pseudo-valuation domains, and conducive domains. Commun. Algebra 30, 1591–1606 (2002)
El-Bast, Z.A., Smith, P.P.: Multiplication modules. Commun. Algebra 16, 755–779 (1988)
Hedstrom, J.R., Houston, E.G.: Pseudo-valuation domains. Pac. J. Math. 75, 137–147 (1978)
Hedstrom, J.R., Houston, E.G.: Pseudo-valuation domains II. Houst. J. Math. 4, 199–207 (1978)
McCasland, R.L., Moore, M.E., Smith, P.F.: An introduction to Zariski spaces over Zariski topologies. Rocky Mt. J. Math. 28, 1357–1369 (1998)
Moghaderi, J., Nekooei, R.: Strongly prime submodules and pseudo-valuation modules. Int. Electron. J. Algebra 10, 65–75 (2011)
Acknowledgments
We would like to thank the referee for the valuable suggestions and comments.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Jahani-Nezhad, R., Khoshayand, F. Pseudo-Valuation Modules. Vietnam J. Math. 44, 477–484 (2016). https://doi.org/10.1007/s10013-015-0165-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10013-015-0165-8