Generalized Discrete Valuation Rings

1969 ◽  
Vol 21 ◽  
pp. 1404-1408 ◽  
Author(s):  
H.-H. Brungs
Keyword(s):  
Noetherian Ring ◽  
Jacobson Radical ◽  
Unit Element ◽  
Global Dimension ◽  
Order Type ◽  
Unique Factorization ◽  
Discrete Valuation Rings ◽  
Valuation Rings ◽  
The Right ◽  
Satisfy Property

Jategaonkar (5) has constructed a class of rings which can be used to provide counterexamples to problems concerning unique factorization in non-commutative domains, the left-right symmetry of the global dimension for a right- Noetherian ring and the transhnite powers of the Jacobson radical of a right- Noetherian ring. These rings have the following property:(W) Every non-empty family of right ideals of the ring R contains exactly one maximal element.In the present paper we wish to consider rings, with unit element, which satisfy property (W). This property means that the right ideals are inverse well-ordered by inclusion, and it is our aim to describe these rings by their order type. Rings of this kind appear as a generalization of discrete valuation rings in R; see (1; 2).In the following, R will always denote a ring with unit element satisfying (W).

Global dimension of factor rings

1994 ◽  
Vol 49 (3) ◽  
pp. 399-411
Author(s):  
Yasuyuki Hirano ◽  
Jae Keol Park ◽  
Klaus W. Roggenkamp
Keyword(s):  
Maximal Ideal ◽  
Noetherian Ring ◽  
Unit Element ◽  
Global Dimension

Let R be a right Noetherian ring with right global dimension bounded by 2, which is integral over its centre, and let a be a regular non-unit element in R. Then R/a; R is right hereditary if and only if a; is not in the square of any maximal ideal of R. More generally, we compare for a right Noetherian ring R which is integral over its center, the global dimension of R with the global dimension of R/(a1R + a2R + … + arR) for a regular R-sequence {ai}, which will allow us to give a considerable extension of a result of Hillman.

scholarly journals Injective homogeneity and the Auslander–Gorenstein property

1995 ◽  
Vol 37 (2) ◽  
pp. 191-204 ◽  
Author(s):  
Zhong Yi
Keyword(s):  
Noetherian Ring ◽  
Projective Dimension ◽  
Global Dimension ◽  
Noetherian Rings ◽  
Connected Component ◽  
Injective Dimension ◽  
The Right

In this paper we refer to [13] and [16] for the basic terminology and properties of Noetherian rings. For example, an FBNring means a fully bounded Noetherian ring [13, p. 132], and a cliqueof a Noetherian ring Rmeans a connected component of the graph of links of R[13, p. 178]. For a ring Rand a right or left R–module Mwe use pr.dim.(M) and inj.dim.(M) to denote its projective dimension and injective dimension respectively. The right global dimension of Ris denoted by r.gl.dim.(R).

scholarly journals Kurosh-Amitsur Right Jacobson Radical of Type 0 for Right Near-Rings

2008 ◽  
Vol 2008 ◽  
pp. 1-6
Author(s):  
Ravi Srinivasa Rao ◽  
K. Siva Prasad ◽  
T. Srinivas
Keyword(s):  
Jacobson Radical ◽  
Paper Properties ◽  
Hereditary Radical ◽  
Constant Part ◽  
The Right

By a near-ring we mean a right near-ring.J0r, the right Jacobson radical of type 0, was introduced for near-rings by the first and second authors. In this paper properties of the radicalJ0rare studied. It is shown thatJ0ris a Kurosh-Amitsur radical (KA-radical) in the variety of all near-ringsR, in which the constant partRcofRis an ideal ofR. So unlike the left Jacobson radicals of types 0 and 1 of near-rings,J0ris a KA-radical in the class of all zero-symmetric near-rings.J0ris nots-hereditary and hence not an ideal-hereditary radical in the class of all zero-symmetric near-rings.

CHAIN CONDITIONS ON NON-SUMMANDS

2011 ◽  
Vol 10 (03) ◽  
pp. 475-489 ◽  
Author(s):  
PINAR AYDOĞDU ◽  
A. ÇIĞDEM ÖZCAN ◽  
PATRICK F. SMITH
Keyword(s):  
Noetherian Ring ◽  
Jacobson Radical ◽  
Finitely Generated ◽  
Chain Conditions ◽  
Finite Dimensional

Let R be a ring. Modules satisfying ascending or descending chain conditions (respectively, acc and dcc) on non-summand submodules belongs to some particular classes [Formula: see text], such as the class of all R-modules, finitely generated, finite-dimensional and cyclic modules, are considered. It is proved that a module M satisfies acc (respectively, dcc) on non-summands if and only if M is semisimple or Noetherian (respectively, Artinian). Over a right Noetherian ring R, a right R-module M satisfies acc on finitely generated non-summands if and only if M satisfies acc on non-summands; a right R-module M satisfies dcc on finitely generated non-summands if and only if M is locally Artinian. Moreover, if a ring R satisfies dcc on cyclic non-summand right ideals, then R is a semiregular ring such that the Jacobson radical J is left t-nilpotent.

scholarly journals On cuspidal representations of general linear groups over discrete valuation rings

2010 ◽  
Vol 175 (1) ◽  
pp. 391-420 ◽  
Author(s):  
Anne-Marie Aubert ◽  
Uri Onn ◽  
Amritanshu Prasad ◽  
Alexander Stasinski
Keyword(s):  
General Linear ◽  
Linear Groups ◽  
General Linear Groups ◽  
Discrete Valuation Rings ◽  
Valuation Rings ◽  
Cuspidal Representations

Right Invariant Right Hereditary Rings

1974 ◽  
Vol 26 (5) ◽  
pp. 1186-1191 ◽  
Author(s):  
H. H. Brungs
Keyword(s):  
Principal Ideal ◽  
Prime Ideals ◽  
Maximal Orders ◽  
Discrete Valuation Rings ◽  
Valuation Rings ◽  
Dedekind Domains ◽  
Left Order ◽  
Principal Ideal Domains

Let R be a right hereditary domain in which all right ideals are two-sided (i.e., R is right invariant). We show that R is the intersection of generalized discrete valuation rings and that every right ideal is the product of prime ideals. This class of rings seems comparable with (and contains) the class of commutative Dedekind domains, but the rings considered here are in general not maximal orders and not Dedekind rings in the terminology of Robson [9]. The left order of a right ideal of such a ring is a ring of the same kind and the class contains right principal ideal domains in which the maximal right ideals are two-sided [6].

Polynomial index in discrete valuation rings

2019 ◽  
Vol 56 (2) ◽  
pp. 260-266
Author(s):  
Mohamed E. Charkani ◽  
Abdulaziz Deajim
Keyword(s):  
Lower Bound ◽  
Valuation Ring ◽  
Irreducible Polynomial ◽  
Discrete Valuation Ring ◽  
Field Extension ◽  
Integral Closure ◽  
Quotient Field ◽  
Discrete Valuation Rings ◽  
Valuation Rings ◽  
Adic Valuation

Abstract Let R be a discrete valuation ring, its nonzero prime ideal, P ∈R[X] a monic irreducible polynomial, and K the quotient field of R. We give in this paper a lower bound for the -adic valuation of the index of P over R in terms of the degrees of the monic irreducible factors of the reduction of P modulo . By localization, the same result holds true over Dedekind rings. As an important immediate application, when the lower bound is greater than zero, we conclude that no root of P generates a power basis for the integral closure of R in the field extension of K defined by P.

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