Mathematics and Statistics Vol. 9(4), pp. 603 - 607
DOI: 10.13189/ms.2021.090419
Reprint (PDF) (264Kb)

The Class of Noetherian RingsWith Finite Valuation Dimension

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

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 valuation 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

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Samsul Arifin , Hanni Garminia , Pudji Astuti , "The Class of Noetherian RingsWith Finite Valuation Dimension," Mathematics and Statistics, Vol. 9, No. 4, pp. 603 - 607, 2021. DOI: 10.13189/ms.2021.090419.

(b). APA Format:
Samsul Arifin , Hanni Garminia , Pudji Astuti (2021). The Class of Noetherian RingsWith Finite Valuation Dimension. Mathematics and Statistics, 9(4), 603 - 607. DOI: 10.13189/ms.2021.090419.