Mathematics and Statistics Vol. 9(4), pp. 521 - 526
DOI: 10.13189/ms.2021.090411
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Finitely Generated Modules's Uniserial Dimensions Over a Discrete Valuation Domain

Samsul Arifin ^1,2,*, Hanni Garminia ¹, Pudji Astuti ^1
1 Algebra Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Jawa Barat, 40132, Indonesia
² Statistics Department, School of Computer Science, Bina Nusantara University, Jakarta, 11480, Indonesia

ABSTRACT

We present some methods for calculating the module's uniserial dimension that finitely generated over a DVD in this article. The idea of a module's uniserial dimension over a commutative ring, which defines how far the module deviates from being uniserial, was recently proposed by Nazemian etc. They show that if R is Noetherian commutative ring, which implies that every finitely generated module over R has uniserial dimension. Ghorbani and Nazemians have shown that R is Noetherian (resp. Artinian) ring if only the ring R X R has (resp. finite) valuation dimension. The finitely generated modules over valuation domain are further examined from here. However, since the region remains too broad, further research into the module's uniserial dimensions that finitely generated over a DVD is needed. In the case of a DVD R, a finitely generated module over R can, as is well-known, be divided into a direct sum of torsion and a free module. Therefore, first, we present methods for determining the primary module's uniserial dimension, and then followed by methods for the general finitely generated module. As can be observed, the module's uniserial dimension is a function of the elementary divisors and the rank of the non torsion module item, which is the major finding of this work.

KEYWORDS
Uniserial Dimension, Valuation Dimension, Finitely Generated, DVD

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Samsul Arifin , Hanni Garminia , Pudji Astuti , "Finitely Generated Modules's Uniserial Dimensions Over a Discrete Valuation Domain," Mathematics and Statistics, Vol. 9, No. 4, pp. 521 - 526, 2021. DOI: 10.13189/ms.2021.090411.

(b). APA Format:
Samsul Arifin , Hanni Garminia , Pudji Astuti (2021). Finitely Generated Modules's Uniserial Dimensions Over a Discrete Valuation Domain. Mathematics and Statistics, 9(4), 521 - 526. DOI: 10.13189/ms.2021.090411.