A note on a noetherian fully bounded ring

TL;DR

This paper proves a theorem relating prime noetherian rings, Gabriel filters, and quotient rings, extending previous results to more general conditions and ordinal dimensions.

Contribution

It introduces a new theorem connecting Gabriel filters, right Ore sets, and quotient rings in prime noetherian rings under weaker hypotheses than prior work.

Findings

Theorem holds under weaker hypotheses than fully bounded rings.

Results extend to rings with ordinal dimension.

The theorem applies to all nonnegative integers m less than n.

Abstract

We prove the following;Theorem:Let R be a prime noetherian ring with k.dimR = n, n a finite non-negative integer. We refer the reader to the definitions (1.1) of this paper.For a fixed non-negative integer m, m<n let Xm be the full set of m-prime ideals p of R and let cm = the set of elements c in R with k-dim(R/cR)< m and let vm = Intersection c(p), for all p in xm.Let c= family of Right ideals I of R such that I intersects cm nontrivially and let v=family of right ideals I of R such that I intersects vm nontrivially.Call g an m-gabriel filter if g=family of Right ideals J of R with k-dim.(R/J)< m.For any simple right module W over any extension ring S of R we denote by r(w) the right annihilator in S of W. Suppose any m critical right R module M with Ass. M = p is such that k.dim. M = R/p = m. Then the following conditions are equivalent:(a) xm has the right intersection condition. (b)(i) g=v.Then vm is a right ore set in R.Let Rv denote the quotient ring of R at the right ore set vm. (ii)Moreover then any simple Rv module say Wv with r(Wv)= qv is a torsion free Rv/qv module. (c)(i)g=c.Then cm is a right ore set in R. Let Rc denote the quotient ring of R at the right ore set cm. (ii) Moreover then any simple Rc module say Wc with r(Wc)=qc is a torsion free Rc/qc module. We may mention that this theorem is proved under a weaker hypothesis on a prime noetherian ring than for a prime noetherian ring that is either fully bounded or has the bijective Gabriel correspondence.In particular the theorem remains true always for these rings for all nonnegative integers m, m<n.Moreover the theorem is true if we replace k-dim. R =n, n finite by any ordinal number.