arXiv subjects

C. L. Wangneo

Publications and source records attributed to C. L. Wangneo.

8 recordsLinked to original sources

arXiv2023

A note on a noetherian fully bounded ring

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.

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arXiv2016

A Note On Noeherian Rings

In this paper we introduce the definition of a noetherian disjoint ring and that of a noetherian non-disjoint ring . For a noetherian ring R , with nilradical N if P and Q represent the semiprime ideals of R called as the right and the left krull-homogenous parts of N as defined in [8] , then we prove the main theorem of this paper for the ring R whose statement is given below. Main Theorem :- Let R be a Noetherian ring with nilradical N . Let P and Q represent the right and the left krull-homogenous parts of N . Then the following hold true for the ring R ; (a) If R is a disjoint ring , then the nilradical N of R is a right and a left weakly ideal invariant ideal of R . Hence N is a right and a left localizable semiprime ideal of R . (b) If R is a non-disjoint ring then the following are equivalent conditions on R ; (i) N is a right and a left weakly ideal invariant ideal of R . (ii) P = Q is a right and a left localizable semiprime ideal of R .

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arXiv2016

On the weak krull symmetry of a noetherian ring

We define when a noetherian ring R is called a right ( or a left) weakly krull symmetric ring . We then prove that if R is a right ( or a left ) krull homogenous ring then R is a right ( or a left ) weakly krull symmetric ring . This result modifies the main result of [2] . The key terms introduced in this paper are of independent interest .

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arXiv2014

On An Application Of The Suslin Monic Polynomial Theorem (I)

In this paper our main theorem states the following, Main Theorem : Let B denote the polynomial ring D[x1,.... ,xn] , in the commuting indeterminates x i over a division ring D . Let M be a finitely generated B-module . Let B m denote the polynomial subring of B , namely D[x1,.... ,xm] , in m , indeterminates , where m is an integer such that 0 ? m ? n , with B0 =D, and Bn =B . Then Krull dimension (M) is m , 0 ? m ? n , if and only if M is a non torsion B m module such that for any positive integer k , k > m , M is a torsion B k module . We then also state and announce a generalisation of the above theorem .

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arXiv2012

On skew polynomial rings

In this note we consider the links of prime ideals of certain skew polynomial rings and prove our main theorem, namely theorem [5], which states the following.Let R be a noetherian ring that is link k-symmetric and let σ be an automorphism of R.Let S(R) denote the skew polynomial ring R[x,σ]. Let B be a prime ideal of S(R) that is extended from R. Then, for a prime ideal D of S(R),there is a link D\rightarrowB in the ring S(R) implies that D is an extended prime ideal of S(R).

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arXiv2012

On links of certain semiprime ideals of a noetherian ring

In this paper we prove our main theorem, namely, theorem (8), which states that a link Q\rightarrowP, of prime ideals Q and P of a noetherian ring R that are σ-semistable with respect to a fixed automorphism σ of R, induces a link Q0\rightarrowP0 of the semiprime ideals Q0 and P0 of the ring R,where Q0 and P0 are the largest σ- invariant or σ- stable ideals contained in the prime ideals Q and P. We also prove a converse to this theorem.

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arXiv2011

On A Certain Krull Symmetry Of a Noetherian Ring

In this note we show that if a noetherian ring R is left and right Krull-homogenous and if: Λ={P\textexclamdown ε spec.R/ |R/P\textexclamdown|_r =|R|_r} and v ={Qj ε spec.R| |R/Qj|l=|R|l} and P =\cap P\textexclamdownε ΛP\textexclamdown and Q = \cap QjεVQj then the following hold true; (1) Pn =0, for some integer n\geq1 (2) Λ=v

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arXiv2011

Prime Ideals in Noetherian Rings

In this short note we study the links of certain prime ideals of a noetherian ring R. We first give the definition of a link krull symmetric noetherian ring R. We then prove theorem 9 that states that for any linked prime ideals P' and Q' of the polynomial ring R[X] where R is a link krull symmetric noetherian ring, if The prime ideal P' is extended then Q' is also an extended prime ideal of R[X]. An application of theorem 9 is then given in theorem 12 for the ring R[X] when R is assumed to be a fully bounded noetherian ring.

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