Strongly Graded Modules and Positively Graded Modules which are Unique Factorization Modules

Yıl 2024, Cilt: 36 Sayı: 36, 1 - 15, 12.07.2024

Iwan Ernanto Indah E. Wijayanti Akira Ueda

https://doi.org/10.24330/ieja.1404435

Öz

Let $M=\oplus_{n\in \mathbb{Z}}M_{n}$ be a strongly graded module over strongly graded ring $D=\oplus_{n\in \mathbb{Z}} D_{n}$. In this paper, we
prove that if $M_{0}$ is a unique factorization module (UFM for short) over $D_{0}$ and $D$ is a unique factorization domain (UFD for short), then $M$ is a UFM over $D$. Furthermore, if $D_{0}$ is a Noetherian domain, we give a necessary and sufficient condition for a positively graded module $L=\oplus_{n\in \mathbb{Z}_{0}}M_{n}$ to be a UFM over positively graded domain $R=\oplus_{n\in \mathbb{Z}_{0}}D_{n}$.

Anahtar Kelimeler

Graded ring, graded module, unique factorization module

Kaynakça

• D. L. Costa, Unique factorization in modules and symmetric algebras, Trans. Amer. Math. Soc., 224(2) (1976), 267-280.
• I. Ernanto, H. Marubayashi, A. Ueda and S. Wahyuni, Positively graded rings which are unique factorization rings, Vietnam J. Math., 49 (2021), 1037-1041.
• I. Ernanto, A. Ueda, I. E. Wijayanti and Sutopo, Some remarks on strongly graded modules, submitted for publication, 2022.
• R. Gilmer, Multiplicative Ideal Theory, Marcel Dekker, New York, 1972.
• C. P. Lu, Factorial modules, Rocky Mountain J. Math., 7 (1977), 125-139.
• H. Marubayashi, S. Wahyuni, I. E. Wijayanti and I. Ernanto, Strongly graded rings which are maximal orders, Sci. Math. Jpn., 82 (2019), 207-210.
• C. Nastasescu and F. van Oystaeyen, Graded Ring Theory, North-Holland Mathematical Library, 28, North-Holland Publishing Co., Amsterdam-New York, 1982.
• A. M. Nicolas, Modules factoriels, Bull. Sci. Math. (2), 95 (1971), 33-52.
• A. M. Nicolas, Extensions factorielles et modules factorables, Bull. Sci. Math. (2), 98 (1974), 117-143.
• M. M. Nurwigantara, I. E. Wijayanti, H. Marubayashi and S. Wahyuni, Krull modules and completely integrally closed modules, J. Algebra Appl., 21(1) (2022), 2350038 (14 pp).
• S. Wahyuni, H. Marubayashi, I. Ernanto and Sutopo, Strongly graded rings which are generalized Dedekind rings, J. Algebra Appl., 19(3) (2020), 2050043 (8 pp).
• S. Wahyuni, H. Marubayashi, I. Ernanto and I. P. Y. Prabhadika, On unique factorization modules: a submodule approach, Axioms, 11(6) (2022), 288 (7 pp).
• I. E. Wijayanti, H. Marubayashi and Sutopo, Positively graded rings which are maximal orders and generalized Dedekind prime rings, J. Algebra Appl., 19(8) (2020), 2050143 (11 pp).
• I. E. Wijayanti, H. Marubayashi, I. Ernanto and Sutopo, Finitely generated torsion-free modules over integrally closed domains, Comm. Algebra, 48(8) (2020), 3597-3607.
• I. E. Wijayanti, H. Marubayashi, I. Ernanto and Sutopo, Arithmetic modules over generalized Dedekind domains, J. Algebra Appl., 21(3) (2022), 2250045 (14 pp).