The Class of Noetherian RingsWith Finite Valuation Dimension

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.