A short Note on prime submodules

Document Type : Research Paper


Department of mathematics, Faculty of sciences, University of Mohaghegh Ardabili, Ardabil, Iran.



Let $R$ be a commutative ring with identity and $M$ be a unital $R$-module. A proper submodule $N$ of $M$ with $N:_RM=\frak p$ is said to be prime or $\frak p$-prime ($\frak p$ a prime ideal of $R$) if $rx\in N$ for $r\in R$ and $x\in M$ implies that either $x\in N$ or $r\in \frak p$. In this paper we study a new equivalent conditions for a minimal prime submodules of an $R$-module to be a finite set, whenever $R$ is a Noetherian ring. Also we introduce the concept of arithmetic rank of a submodule of a Noetherian module and we give an upper bound for it.


[1] S. Abu-Saymeh, On dimensions of nitely generated modules, Comm. Alg. 23(1995), 1131-1144.
[2] J. Azami and M. Khajepour, Topics in prime submodules and other aspects of the prime avoidence theorem, Preprint(Mathematical Reports) .
[3] K. Bahmanpour, A. Khojali and R. Naghipour, A note on minimal prime divisors of an ideal, Algebra Colloq. 18(2011), 727-732.
[4] M. Behboodi, A generalization of the classical Krull dimension for modules, J. Algebra. 305(2006), 1128-1148.
[5] W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge Univ. Press, Cambridge, UK, 1993.
[6] J. Dauns, Prime modules, J. Reine Anegew. Math. 298(1978), 156-181.
[7] J. Jenkins and P. F. Smith, On the prime radical of a module over commutative ring, Comm. Alg. 20(1992), 3593-3602.
[8] O. A. S. Karamzadeh, The Prime Avoidance Lemma revisited, Kyungpook Math. J. 52(2012), 149-153.
[9] C. P. Lu, Unions of prime submodules, Houston J. Math. 23, no.2(1997), 203-213.
[10] K. H. Leung and S. H. Man, On commutative Noetherian rings which satisfy the radical formula, Glasgow math. J. 39(1997), 285-293.
[11] S. H. Man and P. F. Smith, On chains of prime submodules, Israel J. Math. 127(2002), 131-155.
[12] A. Marcelo and J. Munoz Maque, Prime submodules, the discent invariant, and modules of nite length, J. Algebra 189(1997), 273-293.
[13] H. Matsumura, Commutative ring theory, Cambridge Univ. Press, Cambridge, UK, 1986.
[14] R. L. McCasland and P. F. Smith, Prime submodules of Noetherian modules, Rocky Mountain J. Math. 23(1993), 1041-1062.
[15] A. A. Mehrvarz, K. Bahmanpour and R. Naghipour, Arithmetic rank, cohomological dimension and lterregular sequences, J. Alg. Appl. 8(2009), 855-862.
[16] J. J. Rotman, An introduction to homological algebra, Pure Appl. Math., Academic Press, New York, 1979.
[17] D. Pusat-Yilmaz and P. F. Smith, Chain conditions in modules with krull dimension, Comm. Alg. 24(13)(1996), 4123-4133.