Some remarks on generalizations of classical prime submodules

Document Type : Research Paper


Department of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, Semnan, Iran.


Let $R$ be a commutative ring with identity and $M$ be a unitary $R$-module. Suppose that $\phi:S(M)\rightarrow S(M)\cup \lbrace\emptyset\rbrace$ be a function where $S(M)$ is the set of all submodules of $M$. A proper submodule $N$ of $M$ is called an $(n-1, n)$-$\phi$-classical prime submodule, if whenever $r_{1},\ldots,r_{n-1}\in R$ and $m\in M$ with $r_{1}\ldots r_{n-1}m\in N\setminus\phi(N)$, then $r_{1}\ldots r_{i-1}r_{i+1}\ldots r_{n-1}m\in N$, for some $i\in\lbrace 1,\ldots, n-1\rbrace$ $(n\geqslant 3)$.
In this work, $(n-1, n)$-$\phi$-classical prime submodules are studied and some results are established.


[1] Z. Al-Ani, Compactly packed modules and comprimely packed modules, Theses, College of Science, University of Baghdad (1998).
[2] R. Ameri, On the prime submodules of multiplication modules, Inter. J. Math. Math. Sci. 27 (2003),1715-1724.
[3] D. F. Anderson and A. Badawi, On n-absorbing ideals of commutative rings , Comm. Algebra. 39 (2011), 1646-1672.
[4] D. F. Anderson and A. Badawi, On (m, n)-closed ideals of commutative rings, J. Algebra Appl. 16 (1) (2017), 1750013, 21pp.
[5] D. D. Anderson and M. Batanieh, Generalizations of prime ideals, Comm. Algebra. 36 (2008), 686-696.
[6] D. D. Anderson and E. Smith, Weakly prime ideals, Houston J. Math. 29 (2003), 831-840.
[7] A. Ashour, Primary Finitely Compactly Packed Modules and S-Avoidance Theorem for Modules, Turkish J. Math. 32 (2008), 315-324.
[8] A. Azizi, Prime submodules and at modules, Acta Math. Sin. ( Engl. Ser.) 23 (1) (2007). 147-152.
[9] A. Azizi, Weakly prime submodules and prime submodules, Glasg. Math. J. 48 (2) (2006), 343-346.
[10] A. Y. Darani and F. Soheilnia, On n-absorbing submdules, Math. Commun. 17 (2012), 547-557.
[11] J. Dauns, Prime submodules , J. Reine Angew. Math. 298 (1978) 156-181.
[12] M. Ebrahimpour and F. Mirzaee, On φ-Semiprime Submodules, J. Korean Math. Soc. 54, No. 4 (2017)1099-1108.
[13] M. Ebrahimpour and R. Nekooei, On generalizations of prime submodules, Bull. Iranian Math. Soc. 39 (5)(2013), 919-939.
[14] Z. A. El-Bast and P.F.Smith, Multiplication modules, Comm. Algebra. 16 (1988), 766-779.
[15] A. K. Jabbar, A generalization of prime and weakly prime submodules, Pure Math. Sci. 2 (2013), 1-11.
[16] C. P. Lu, Prime submodules of modules, Comment. Math. Univ. St. Pauli 33 (1984), 61-69.
[17] C. P. Lu, Spectra of modules, Comm. Algebra. 23 (1995), 3741-3752.
[18] Marcelo, A., Masque, M., Prime submodule, the descent invariant, and module of nite length, J. Algebra.
189 (1997), 273-293.
[19] R. L. McCasland and M.E. Moore, Prime submodules, Comm. Algebra. 20 (1992), 1803-1817.
[20] H. Mostafanasab, E.S. Sevim, S. Babaei and U.Tekir, φ-Classical prime submodules, arXiv:1507.08981v1 [math.AC] 30 Jul 2015.
[21] J. V. Pakala and T.S. Shores, On Compactly Packed Rings, Paci c J. Math. Vol. 97, No. 1 (1981), 197-201.
[22] P. Quartararo and H.S. Butts, Finite unions of ideals and modules, Proc. Amer. Math. Soc. 52 (1975), 91-96.
[23] P.F. Smith, Some remarks on multiplication modules, Arch. Math. 50 (1988), 223-235.
[24] N. Zamani, φ-prime submodules, Glasg. Math. J. 52(2)(2010), 253-259.