Document Type : Research Paper

**Author**

Department of Mathematics, Mobarakeh Branch, Islamic Azad University, Isfahan, Iran,

**Abstract**

Let $R$ be a commutative ring with identity and $M$ be a unitary $R$-module. First, we study multiplication $R$-modules $M$ where $R$ is a one dimensional Noetherian ring or $M$ is a finitely generated $R$-module. In fact, it is proved that if $M$ is a multiplication $R$-module over a one dimensional Noetherian ring $R$, then $M\cong I$ for some invertible ideal $I$ of $R$ or $M$ is cyclic. Also, a multiplication $R$-module $M$ is finitely generated if and only if $M$ contains a finitely generated submodule $N$ such that $Ann_R(N)= Ann_R(M)$. A submodule $N$ of $M$ is called dense in $M$, if $M=\sum_\varphi\varphi(N)$ where $\varphi$ runs over all the $R$-homomorphisms from $N$ into $M$ and $R$-module $M$ is called a weak $\pi$-module if every non-zero finitely generated submodule is dense in $M$. It is shown that a faithful multiplication module over an integral domain $R$ is a weak $\pi$-module if and only if it is a Prufer prime module.

**Keywords**

[1] M.M. Ali and D.J. Smith, Some remarks on multiplication and projective modules, Communications in Algebra, Vol. 32 No. 10 (2004), pp. 3897–3909.

[2] M. Alkan and Y. Tras, On invertible and dense submodules, Communications in Algebra, Vol. 32 No. 10 (2004), pp. 3911–3919.

[3] A. Azizi, Principal ideal multiplication modules, Algebra Colloquium, Vol. 15 No. 04 (2008), pp. 637–648.

[4] A. Barnard, Multiplication modules, Journal of Algebra, Vol. 71 (1981), pp. 174–178.

[5] M. Behboodi, On prime modules and dense submodules, Journal of commutative Algebra, Vol. 4 No. 4 (2012), pp. 479–488.

[6] Z.A. El-Bast and P.F. Smith, Multiplication modules, Communications in Algebra, Vol. 16 (1988), pp. 755–799.

[7] G.D. Findlay and J. Lambek, A generalized ring of quotients, I, II, Canadian Mathematical Bulletin, Vol. 1 (1958), pp. 77–85, 155–167.

[8] A. Hajikarimi and A.R. Naghipour, A note on monoform modules, Bulletin of the Korean Mathematical Society, Vol. 56 No. 2 (2019), pp. 505–514.

[9] A.G. Naoum and F.G. Al-Alwan, Dense submodules of multiplication modules, Communications in Algebra, Vol. 24 No. 2 (1996), pp. 413–424.

[10] A.G. Naoum and F.G. Al-Alwan, Dedekind modules, Communications in Algebra, Vol. 24 No. 2 (1996), pp. 397–412.

[11] W.K. Nicholson, J.K. Park and M.F. Yousif, Principally quasi-injective modules, Communications in Algebra, Vol. 27 No. 4 (1999), pp. 1683–1693.

[12] P.F. Smith, Some remarks on multiplication modules, Archiv der Mathematik, Vol. 50 (1988), pp. 223–235.

February 2021

Pages 89-97