Researchers

Shawki Al Rashed

Published in

Al-Baath University Journal, Volume 44, Issue 13, December 2022.

Abstract

This paper is an extension and follow-up to the two papers [2] and [1]. It presents the relationship between the Arithmetical Rings and the Normal Rings. A general idea of the importance of arithmetic rings and normal rings was presented in the introduction, and basic definitions and concepts of commutative algebra in the first paragraph, and in the second paragraph, was presented the relationship between Pruefer's Domain and locally normal ring as in [2], and the Description of arithmetic rings using maximal ideals and Pruefer's Domain as in [1]. In the third paragraph, we show that every arithmetic ring is normal in case it is integral domain, but the opposite is not necessarily true. Then, it is proposed to prove that the associated minimum prime ideals are the same as the minimum prime ideals by proving that the set of embedded associated prime ideals is equal to an empty set, and that their number is limited, that in reduced noether rings, Within these conditions, we have shown that the minimum prime ideals exist, their number is bounded, and they are prime among themselves, two by two, and that the remainder ring (division) of a reduced noether ring to a minimum prime ideal is normal if and only if the ring is arithmetic.

Keywords: Primary Decomposition, Arithmetical Ring, Normal Ring, Noether's Ring, Krull's Dimension.

Link to full issue

https://albaath-univ.edu.sy/journal/index.php/BUJB/issue/view/452/407