On strongly dense submodules‎

Document Type : Research Paper


Department of Mathematics‎, ‎Shahid Chamran University of Ahvaz‎, ‎Ahvaz‎, ‎Iran.


The submodules with the property of the title ( a submodule $N$ of an $R$-module $M$ is called strongly dense in $M$, denoted by $N\leq_{sd}M$, if for any index set $I$, $\prod _{I}N\leq_{d}\prod _{I}M$) are introduced and fully investigated. It is shown that for each submodule $N$ of $M$ there exists the smallest subset $D'\subseteq M$ such that $N+D'$ is a strongly dense submodule of $M$ and $D'\bigcap N=0$. We also introduce a class of modules in which the two concepts of strong essentiality and strong density coincide. It is also shown that for any module $M$, dense submodules in $M$ are strongly dense if and only if $M\leq_{sd} \tilde{E}(M)$, where $\tilde{E}(M)$ is the rational hull of $M$. It is proved that $R$ has no strongly dense left ideal if and only if no nonzero-element of every cyclic $R$-module $M$ has a strongly dense annihilator in $R$. Finally, some appropriate properties and new concepts related to strong density are defined and studied.


Main Subjects

E. P. Armendariz, Rings with dcc on essential left ideals, Comm. Algebra. 8 (1980), no. 3, 299--308.
M. Behboodi, O. A. S. Karamzadeh and H. Koohy, Modules whose certain submodules are prime, Vietnam J. Math. 32 (2004), no. 3, 303--317.
G. D. Findlay and J. Lambek, A generalized ring of quotients, I, II, Canad. Math. Bull. 1 (1958), 77--85, 155--167.
M. Ghirati and O. A. S. Karamzadeh, On strongly essential submodules, Comm. Algebra. 36 (2008), no. 2, 564--580.
K. R. Goodearl and R. Jr. Warfield, An introduction to noncommutative Noetherian rings, second edition, Cambridge University Press, Cambridge, 2004.
O. A. S. Karamzadeh, M. Motamedi and S. M. Shahrtash, On rings with a unique proper essential right ideal, Fund. Math. 183 (2004), no. 3, 229--244.
T. Y. Lam, Lectures on Modules and Rings, Graduate Texts in Math., Springer-Verlag, 189, Berlin-Heidelberg-New York, 1999.
T. Y. Lam, Exercises in Modules and Rings, Problem Books in Math., Springer, New York, 2007.
J. Lambek and G. Michler, The torsion theory at a prime ideal of a right Noetherian ring, J. Algebra 25 (1973), no. 2, 364--389.
K. Louden, Maximal quotient rings of ring extensions, Pacific J. Math. 62 (1976), no. 2, 489--496.
L. B. Stenstrom, Rings of quotients, Die Grundlehren der Mathematischen Wissenschaften, Band 217. An introduction to methods of ring theory, Springer-Verlag, New York-Heidelberg, 1975.
H. H. Storrer, Goldman's primary decomposition, Lectures on ring and modules, Lecture Notes in Math., 246, Springer, Berlin, 1972.
J. M. Zelmanowitz, Representation of rings with faithful polyform modules, Comm. Algebra. 14 (1986), no. 6, 1141--1169.