scholarly journals Unique factorisation rings

1992 ◽  
Vol 35 (2) ◽  
pp. 255-269 ◽  
Author(s):  
A. W. Chatters ◽  
M. P. Gilchrist ◽  
D. Wilson
Keyword(s):  
Prime Ideal ◽  
Prime Ring ◽  
Polynomial Identity ◽  
Noetherian Ring ◽  
Basic Theory ◽  
Maximal Order ◽  
Prime Element ◽  
Polynomial Extension ◽  
Simple Ring ◽  
Left And Right

Let R be a ring. An element p of R is a prime element if pR = Rp is a prime ideal of R. A prime ring R is said to be a Unique Factorisation Ring if every non-zero prime ideal contains a prime element. This paper develops the basic theory of U.F.R.s. We show that every polynomial extension in central indeterminates of a U.F.R. is a U.F.R. We consider in more detail the case when a U.F.R. is either Noetherian or satisfies a polynomial identity. In particular we show that such a ring R is a maximal order, that every height-1 prime ideal of R has a classical localisation in which every two-sided ideal is principal, and that R is the intersection of a left and right Noetherian ring and a simple ring.

scholarly journals One sided invertibility and localisation

1992 ◽  
Vol 34 (3) ◽  
pp. 333-339 ◽  
Author(s):  
C. R. Hajarnavis
Keyword(s):  
Prime Ideal ◽  
Maximal Ideal ◽  
Polynomial Identity ◽  
Noetherian Ring ◽  
Crucial Role ◽  
Prime Ideals ◽  
Careful Study ◽  
Definition Of ◽  
The Right

In general, a prime ideal P of a prime Noetherian ring need not be classically localisable. Since such a localisation, when it does exist, is a striking property; sufficiency criteria which guarantee it are worthy of careful study. One such condition which ensures localisation is when P is an invertible ideal [5, Theorem 1.3]. The known proofs of this result utilise both the left as well as the right invertiblity of P. Such a requirement is, in practice, somewhat restrictive. There are many occasions such as when a product of prime ideals is invertible [6] or when a non-idempotent maximal ideal is known to be projective only on one side [2], when the assumptions lead to invertibilty also on just one side. Our main purpose here is to show that in the context of Noetherian prime polynomial identity rings, this one-sided assumption is enough to ensure classical localisation [Theorem 3.5]. Consequently, if a maximal ideal in such a ring is invertible on one side then it is invertible on both sides [Proposition 4.1]. This result plays a crucial role in [2]. As a further application we show that for polynomial identity rings the definition of a unique factorisation ring is left-right symmetric [Theorem 4.4].

scholarly journals Polycyclic group rings and unique factorisation rings

1994 ◽  
Vol 36 (2) ◽  
pp. 135-144
Author(s):  
Kenneth W. MacKenzie
Keyword(s):  
Prime Ideal ◽  
Noetherian Ring ◽  
Commutative Rings ◽  
Prime Element ◽  
Noetherian Rings ◽  
Group Rings ◽  
Noetherian Domain ◽  
Noncommutative Version ◽  
Normal Element ◽  
Completely Prime Ideal

The theory of unique factorisation in commutative rings has recently been extended to noncommutative Noetherian rings in several ways. Recall that an element x of a ring R is said to be normalif xR = Rx. We will say that an element p of a ring R is (completely) prime if p is a nonzero normal element of R and pR is a (completely) prime ideal. In [2], a Noetherian unique factorisation domain (or Noetherian UFD) is defined to be a Noetherian domain in which every nonzero prime ideal contains a completely prime element: this concept is generalised in [4], where a Noetherian unique factorisation ring(or Noetherian UFR) is defined as a prime Noetherian ring in which every nonzero prime ideal contains a nonzero prime element; note that it follows from the noncommutative version of the Principal Ideal Theorem that in a Noetherian UFR, if pis a prime element then the height of the prime ideal pR must be equal to 1. Surprisingly many classes of noncommutative Noetherian rings are known to be UFDs or UFRs: see [2] and [4] for details. This theory has recently been extended still further, to cover certain classes of non-Noetherian rings: see [3].

scholarly journals A note on Noetherian orders in Artinian rings

1979 ◽  
Vol 20 (2) ◽  
pp. 125-128 ◽  
Author(s):  
A. W. Chatters
Keyword(s):  
Noetherian Ring ◽  
Quotient Ring ◽  
Triangular Matrix ◽  
Idempotent Element ◽  
Noetherian Rings ◽  
Central Idempotent ◽  
Matrix Rings ◽  
Regular Elements ◽  
Zero Divisors ◽  
Left And Right

Throughout this note, rings are associative with identity element but are not necessarily commutative. Let R be a left and right Noetherian ring which has an Artinian (classical) quotient ring. It was shown by S. M. Ginn and P. B. Moss [2, Theorem 10] that there is a central idempotent element e of R such that eR is the largest Artinian ideal of R. We shall extend this result, using a different method of proof, to show that the idempotent e is also related to the socle of R/N (where N, throughout, denotes the largest nilpotent ideal of R) and to the intersection of all the principal right (or left) ideals of R generated by regular elements (i.e. by elements which are not zero-divisors). There are many examples of left and right Noetherian rings with Artinian quotient rings, e.g. commutative Noetherian rings in which all the associated primes of zero are minimal together with full or triangular matrix rings over such rings. It was shown by L. W. Small that if R is any left and right Noetherian ring then R has an Artinian quotient ring if and only if the regular elements of R are precisely the elements c of R such that c + N is a regular element of R/N (for further details and examples see [5] and [6]). By the largest Artinian ideal of R we mean the sum of all the Artinian right ideals of R, and it was shown by T. H. Lenagan in [3] that this coincides in any left and right Noetherian ring R with the sum of all the Artinian left ideals of R.

scholarly journals A note on the invertible ideal theorem

1984 ◽  
Vol 25 (1) ◽  
pp. 27-30 ◽  
Author(s):  
Andy J. Gray
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Simple Proof ◽  
Noetherian Ring ◽  
Principal Ideal ◽  
Central Element ◽  
Commutative Case ◽  
Additional Information ◽  
The Subject

This note is devoted to giving a conceptually simple proof of the Invertible Ideal Theorem [1, Theorem 4·6], namely that a prime ideal of a right Noetherian ring R minimal over an invertible ideal has rank at most one. In the commutative case this result may be easily deduced from the Principal Ideal Theorem by localizing and observing that an invertible ideal of a local ring is principal. Our proof is partially analogous in that it utilizes the Rees ring (denned below) in order to reduce the theorem to the case of a prime ideal minimal over an ideal generated by a single central element, which can be easily dealt with by adapting the commutative argument in [8]. The reader is also referred to the papers of Jategaonkar on the subject [5, 6, 7], particularly the last where another proof of the theorem appears which yields some additional information.

On Duality Relative to Self-orthogonal Bimodules

2005 ◽  
Vol 12 (02) ◽  
pp. 319-332
Author(s):  
Jun Wu ◽  
Wenting Tong
Keyword(s):  
Noetherian Ring ◽  
Injective Dimension ◽  
Left And Right

Let R be a left and right Noetherian ring and RωR a faithfully balanced self-orthogonal bimodule. We introduce the notions of special embeddings and modules of ω-D-class n, and then give some characterizations of them. As an application, we study the properties of RωR with finite injective dimension. Our results extend the main results in [4].

On the Ring of Quotients at a Prime Ideal of a Right Noetherian Ring

1972 ◽  
Vol 24 (4) ◽  
pp. 703-712 ◽  
Author(s):  
A. G. Heinicke
Keyword(s):  
Prime Ideal ◽  
Noetherian Ring ◽  
Quotient Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Torsion Class ◽  
Ring Of Quotients ◽  
Ore Condition ◽  
The Right ◽  
Necessary And Sufficient

J. Lambek and G. Michler [3] have initiated the study of a ring of quotients RP associated with a two-sided prime ideal P in a right noetherian ring R. The ring RP is the quotient ring (in the sense of [1]) associated with the hereditary torsion class τ consisting of all right R-modules M for which HomR(M, ER(R/P)) = 0, where ER(X) is the injective hull of the R-module X.In the present paper, we shall study further the properties of the ring RP. The main results are Theorems 4.3 and 4.6. Theorem 4.3 gives necessary and sufficient conditions for the torsion class associated with P to have property (T), as well as some properties of RP when these conditions are indeed satisfied, while Theorem 4.6 gives necessary and sufficient conditions for R to satisfy the right Ore condition with respect to (P).

Positive deissler rank and the complexity of injective modules

1988 ◽  
Vol 53 (1) ◽  
pp. 284-293 ◽  
Author(s):  
T. G. Kucera
Keyword(s):  
Prime Ideal ◽  
Lower Bounds ◽  
Upper Bound ◽  
Noetherian Ring ◽  
Krull Dimension ◽  
Primary Ideal ◽  
Prime Ideals ◽  
Commutative Noetherian Ring ◽  
Injective Modules ◽  
Primary Ideals

This is the second of two papers based on Chapter V of the author's Ph.D. thesis [K1]. For acknowledgements please refer to [K3]. In this paper I apply some of the ideas and techniques introduced in [K3] to the study of a very specific example. I obtain an upper bound for the positive Deissler rank of an injective module over a commutative Noetherian ring in terms of Krull dimension. The problem of finding lower bounds is vastly more difficult and is not addressed here, although I make a few comments and a conjecture at the end.For notation, terminology and definitions, I refer the reader to [K3]. I also use material on ideals and injective modules from [N] and [Ma]. Deissler's rank was introduced in [D].For the most part, in this paper all modules are unitary left modules over a commutative Noetherian ring Λ but in §1 I begin with a few useful results on totally transcendental modules and the relation between Deissler's rank rk and rk+.Recall that if P is a prime ideal of Λ, then an ideal I of Λ is P-primary if I ⊂ P, λ ∈ P implies that λn ∈ I for some n and if λµ ∈ I, λ ∉ P, then µ ∈ I. The intersection of finitely many P-primary ideals is again P-primary, and any P-primary ideal can be written as the intersection of finitely many irreducible P-primary ideals since Λ is Noetherian. Every irreducible ideal is P-primary for some prime ideal P. In addition, it will be important to recall that if P and Q are prime ideals, I is P-primary, J is Q-primary, and J ⊃ I, then Q ⊃ P. (All of these results can be found in [N].)

On Prime Rings with Ascending Chain Condition on Annihilator Right Ideals and Nonzero Infective Right Ideals

1971 ◽  
Vol 14 (3) ◽  
pp. 443-444 ◽  
Author(s):  
Kwangil Koh ◽  
A. C. Mewborn
Keyword(s):  
Prime Ring ◽  
Chain Condition ◽  
Prime Rings ◽  
Simple Ring ◽  
Descending Chain Condition ◽  
Image Position ◽  
The Right ◽  
Ascending Chain Condition

If I is a right ideal of a ring R, I is said to be an annihilator right ideal provided that there is a subset S in R such thatI is said to be injective if it is injective as a submodule of the right regular R-module RR. The purpose of this note is to prove that a prime ring R (not necessarily with 1) which satisfies the ascending chain condition on annihilator right ideals is a simple ring with descending chain condition on one sided ideals if R contains a nonzero right ideal which is injective.

Derivations with annihilator conditions on Lie ideals in prime rings

2019 ◽  
Vol 19 (02) ◽  
pp. 2050025 ◽  
Author(s):  
Shuliang Huang
Keyword(s):  
Prime Ring ◽  
Polynomial Identity ◽  
Lie Ideal ◽  
Prime Rings ◽  
Positive Integers

Let [Formula: see text] be a prime ring with characteristic different from two, [Formula: see text] a derivation of [Formula: see text], [Formula: see text] a noncentral Lie ideal of [Formula: see text], and [Formula: see text]. In the present paper, it is shown that if one of the following conditions holds: (i) [Formula: see text], (ii) [Formula: see text], (iii) [Formula: see text] and (iv) [Formula: see text] for all [Formula: see text], where [Formula: see text] are fixed positive integers, then [Formula: see text] unless [Formula: see text] satisfies [Formula: see text], the standard polynomial identity in four variables.

scholarly journals On n-generalized commutators and Lie ideals of rings

2021 ◽  
pp. 2250221
Author(s):  
Peter V. Danchev ◽  
Tsiu-Kwen Lee
Keyword(s):  
Positive Integer ◽  
Prime Ring ◽  
Classical Result ◽  
Associative Ring ◽  
Nonzero Ideal ◽  
Lie Ideal ◽  
Additive Subgroup ◽  
Simple Ring ◽  
Noncommutative Polynomials ◽  
Generalized Commutator

Let [Formula: see text] be an associative ring. Given a positive integer [Formula: see text], for [Formula: see text] we define [Formula: see text], the [Formula: see text]-generalized commutator of [Formula: see text]. By an [Formula: see text]-generalized Lie ideal of [Formula: see text] (at the [Formula: see text]th position with [Formula: see text]) we mean an additive subgroup [Formula: see text] of [Formula: see text] satisfying [Formula: see text] for all [Formula: see text] and all [Formula: see text], where [Formula: see text]. In the paper, we study [Formula: see text]-generalized commutators of rings and prove that if [Formula: see text] is a noncommutative prime ring and [Formula: see text], then every nonzero [Formula: see text]-generalized Lie ideal of [Formula: see text] contains a nonzero ideal. Therefore, if [Formula: see text] is a noncommutative simple ring, then [Formula: see text]. This extends a classical result due to Herstein [Generalized commutators in rings, Portugal. Math. 13 (1954) 137–139]. Some generalizations and related questions on [Formula: see text]-generalized commutators and their relationship with noncommutative polynomials are also discussed.

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