The canonical form of multiplication modules
DOI :
https://doi.org/10.5269/bspm.52858Résumé
Let $R$ be a commutative ring with unit. An $R$-module $M$ is called a multiplication module if for every submodule $N$ of $M$, there is an ideal $I$ of $R$ such that $N=IM$. $M$ is called also a CF-module if there is some ideals $I_1,...,I_n$ of $R$ such that $M \simeq R/I_1 \bigoplus R/I_2 \bigoplus ... \bigoplus R/I_n$ and $I_1 \subseteq I_2 \subseteq ... \subseteq I_n$. In this paper, we use some new results about $\mu_R(M)$ the minimal number of generators of $M$ to show that a finitely generated multiplication module is a CF-module if and only if it is a cyclic module.Références
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2. Azizi, A., Shiraz, Weak multiplication modules. Czechoslovak mathematical journal 53(128), 529-534, (2003)
3. Lang, S., Algebra. 3rd Edition, Addison-Wesley, (1993).
4. Kaplansky, I., Elementary Divisors and Modules. Tran. Ame. Math. Soc. 66, 153-169, (1949).
5. M. E. Charkani, M. E., Akharraz, I., Fitting ideals and cyclic decomposition of finitely generated modules. Arabian Journal for Science and Engineering 25(2), 151-156, (2000).
6. Cayley, A., On the theory of involution in geometry,Cambridge Math. 11, 52-61, (1847).
7. Lombardi, H., Quitte,C., Algebre Commutative, Methodes Constructives: Modules Projectifs de Type Fini, Calvage et Mounet, (2016).
8. Gilmer, R., Heinzer, W., On the Number of Generators of an Invertible Ideal, Journal of Algebra 14, 139-151, (1970).
9. Kumar, N. M., On Two Conjectures About Polynomial Rings, Inventiones Math. 46, 225-236, (1978).
10. Brown, W. C., Matrices over commutative rings. Marcel Decker Inc, New York, 149-175, (1993).
11. Shores, T., Wiegand, R., Rings whose finitely generated modules are direct sums of cyclics. Journal of Algebra 32, 152-172, (1974).
12. Lafond, J. P., Anneaux locaux commutatifs sur lesquels tout module de type fini est somme directe de modules monog`enes. Journal of Algebra 17, 575 - 591, (1971).
13. Barnard, A., Multiplication modules. Journal of Algebra 71, 174-178, (1981).
14. Eisenbud, D., Commutative Algebra with a view towards Algebraic Geometry. GTM 150, Springer Verlag, New York (1994).
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2022-12-24
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