Document Type : Original Manuscript

Author

Department of Mathematics, Faculty of Science, Razi University, Kermanshah, 67149-67346, Iran.

Abstract

Let $R$ be a commutative ring with identity and $M$ be a unitary $R$-module. An $R$-module $M$ is called a multiplication module if for every submodule $N$ of $M$ there exists an ideal $I$ of $R$ such that $N = IM$. It is shown that over a Noetherian domain $R$ with dim$(R)\leq 1$, multiplication modules are precisely cyclic or isomorphic to an invertible ideal of $R$. Moreover, we give a characterization of finitely generated multiplication modules.

Keywords

###### ##### References
1. T. Alsuraiheed, and V. V. Bavula, Characterization of multiplication commutative rings with finitely many minimal prime ideals, Comm. Algebra, 47 (2019), 1–8.
2. A. Azizi, and C. Jayaram, On principal ideal multiplication modules, Ukrain. Math. J., 69 (2017), 337–347.
3. A. Barnard, Multiplication modules, J. Algebra, 71 (1981), 174–178.
4. Z. A. El-Bast, and P. P. Smith, Multiplication modules, Comm. Algebra, 16 (1988), 755–779.
5. A. Naoum, Flat modules and multiplication modules, Period. Math. Hungar., 21 (1990), 309–317.
6. J. C. Perez, J. R. Montes and G. T. Sanchez, A generalization of multiplication modules, Bull. Korean Math. Soc., 56 (2019), 83–102.
7. T. Y. Lam, Lectures on Modules and Rings, Vol. 189. Springer Science & Business Media, 2012.
8. P. F. Smith, Fully invariant multiplication modules, Palestine J. Math., 4 (2015),
462–470.
9. A. A. Tuganbaev, Multiplication modules over non-commutative rings, Sbornik: Mathematics, 194 (2003), 1837–1864.
10. A. A. Tuganbaev, Flat and multiplication modules, J. Math. Sci. (N. Y.) 128 (2005), 2998–3004.