Mathematics and Statistics 9(4): 603-607, 2021 DOI: 10.13189/ms.2021.090419 http://www.hrpub.org The Class of Noetherian Rings With Finite Valuation Dimension 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 Received March 8, 2021; Revised June 14, 2021; Accepted July 19, 2021 Cite This Paper in the following Citation Styles (a): [1] Samsul Arifin, Hanni Garminia, Pudji Astuti, ”The Class of Noetherian Rings With Finite Valuation Dimension,” Mathematics and Statistics, Vol.9, No.4, pp. 603-607, 2021. DOI: 10.13189/ms.2021.090419 (b): Samsul Arifin, Hanni Garminia, Pudji Astuti, (2021). The Class of Noetherian Rings With Finite Valuation Dimension. Mathematics and Statistics, 9(4), 603-607. DOI: 10.13189/ms.2021.090419 Copyright ©2021 by authors, all rights reserved. Authors agree that this article remains permanently open access under the terms of the Creative Commons Attribution License 4.0 International License Abstract Not a long time ago, Ghorbani and Nazemian [2015] introduced the concept of dimension of valuation which measures how much does the ring differ from the valuation. They’ve shown that every Artinian ring has a finite valuation dimensions. Further, any comutative ring with a finite valua- tion dimension is semiperfect. However, there is a semiperfect ring which has an infinite valuation dimension. With those facts, it is of interest to further investigate property of rings that has a finite dimension of valuation. In this article we define conditions that a Noetherian ring requires and suffices to have a finite valuation dimension. In particular we prove that, if and only if it is Artinian or valuation, a Noetherian ring has its finite valuation dimension. In view of the fact that a ring needs a semi perfect dimension in terms of valuation, our investigation is confined on semiperfect Noetherian rings. Furthermore, as a finite product of local rings is a semi perfect ring, the inquiry into our outcome is divided into two cases, the case of the examined ring being local and the case where the investigated ring is a product of at least two local rings. This is, first of all, that every local Noetherian ring possesses a finite valuation dimension, if and only if it is Artinian or valuation. Secondly, any Notherian Ring generated by two or more local rings is shown to have a finite valuation dimension, if and only if it is an Artinian. Keywords Uniserial Dimension, Dimension of Valuation, Local Rings, Semiperfect Rings 1 Introduction A valuation ring can be recognized by collecting any ideals fully arranged by the inclusion relationship. The concept of the valuation ring has been widely studied, for examples, in [7], [8], [12], and [13]. The measure that states the dimension of the ring is from the valuation condition is called dimension of valuation, and we can see this notion the first time in [9]. By definition, the dimension of valuation of a ring is the uniserial dimension from its group over itself. We can see the notion of uniserial dimension the first time in [15]. The properties of rings that have a dimension of valuation and groups with uniserial dimension have been studied (see [15]). In another literature, Ghorbani and Nazemian [9] show that any Artinian ring has a finite dimension of valuation. Ar- tinian rings, on the other hand, are not all comutative rings with a finite valuation dimension. Moreover, any rings with a finite valuation dimension is considered semiperfect (see [9]). But the converse is not always true, since there exists a semiperfect Noetherian ring with infinite dimension of valuation. The foregoing facts make one wonder what conditions com- mutative rings must meet in order to have a finite valuation dimension. As part of an effort to answer that question, in this note we prove that the class containing Noetherian rings that have finite dimension of valuation consists of all Artinian rings and valuation rings. In Section 2, we start with some review of the concept of the dimension of valuation. Our main result is the characterization of Noetherian rings that has a finite di- mension of valuation in Section 3.