Gabriel Quotient Rings

TL;DR

This paper introduces the concept of m-Gabriel quotient rings for prime Noetherian rings, extending the classical quotient ring construction with properties related to prime ideals and Krull dimension.

Contribution

It defines and constructs the m-Gabriel quotient ring, generalizing Gabriel quotient rings by incorporating Krull dimension considerations and prime ideal structures.

Findings

Construction of R(m) as a subring of the Goldie quotient ring Q

R(m) has units corresponding to the set c_m of elements with Krull dimension less than m

R(m) forms an Ore set for the set of m-prime ideals

Abstract

In this paper we prove the following theorem. Let R be a prime Noetherian ring with krull dimension |R| = n where n is a positive integer. Let Q be the Goldie quotient ring of R. For a fixed positive integer m < n, let xm be the set of all prime ideals of R such that krull dimension R/p equals m. Call xm the set of m-full prime ideals of R. Let Cm be the set of elements c of R With krull dimension R/cR less than m. Call g as the m-gabriel filter, if g is the family of right ideals I of R with krull dimension R/I less than m. We construct an extension ring R(m) of R having the following properties (i) R(m) is a subring of Q with identity element 1 of R. (ii) If u(R(m)) is the set of units of R(m) then u(R(m) intersection R equals the set cm. (iii) For a full set of m-prime ideals of R the set cm is a right ore set of R . We call R(m) as the m-Gabriel quotient ring of R.