scholarly journals MÜLLER’S QUESTION ON SEMI-PERFECT COMPLETE HEREDITARY NOETHERIAN PRIME RINGS

2006 ◽  
Vol 49 (3) ◽  
pp. 567-573
Author(s):  
Pham Ngoc Ánh ◽  
Dolors Herbera
Keyword(s):  
Prime Ring ◽  
Positive Answer ◽  
Prime Rings ◽  
Self Duality ◽  
Hereditary Noetherian Prime Ring ◽  
Weakly Symmetric

AbstractA positive answer to a question of Müller is given: any semi-perfect complete hereditary Noetherian prime ring $R$ has a weakly symmetric self-duality sending every ideal $I$ to its cycle-neighbour $X$. Consequently, the factor rings $R/I$ and $R/X$ are isomorphic without using the 1984 results of Dischinger and Müller.

Rings with Enough Invertible Ideals

1983 ◽  
Vol 35 (1) ◽  
pp. 131-144 ◽  
Author(s):  
P. F. Smith
Keyword(s):  
Prime Ring ◽  
Regular Element ◽  
Elementary Proof ◽  
Identity Element ◽  
Quotient Ring ◽  
Minimum Condition ◽  
Prime Rings ◽  
Hereditary Noetherian Prime Ring ◽  
The Right

All rings are associative with identity element 1 and all modules are unital. A ring has enough invertible ideals if every ideal containing a regular element contains an invertible ideal. Lenagan [8, Theorem 3.3] has shown that right bounded hereditary Noetherian prime rings have enough invertible ideals. The proof is quite ingenious and involves the theory of cycles developed by Eisenbud and Robson in [5] and a theorem which shows that any ring S such that R ⊆ S ⊆ Q satisfies the right restricted minimum condition, where Q is the classical quotient ring of R. In Section 1 we give an elementary proof of Lenagan's theorem based on another result of Eisenbud and Robson, namely every ideal of a hereditary Noetherian prime ring can be expressed as the product of an invertible ideal and an eventually idempotent ideal (see [5, Theorem 4.2]). We also take the opportunity to weaken the conditions on the ring R.

Modules Over Hereditary Noetherian Prime Rings, II

1976 ◽  
Vol 28 (1) ◽  
pp. 73-82 ◽  
Author(s):  
Surjeet Singh
Keyword(s):  
Prime Ring ◽  
Projective Modules ◽  
Prime Rings ◽  
Hereditary Noetherian Prime Ring ◽  
Torsion Modules

Let R be a hereditary noetherian prime ring ((hnp)-ring) with enough invertible ideals. Torsion modules over bounded (hnp)-rings were studied by the author in [10; 11]. All the results proved in [10; 11] also hold for torsion R-modules having no completely faithful submodules. In Section 2, indecomposable injective torsion R-modules which are not completely faithful are studied, and they are shown to have finite periodicities (Theorem (2.8) and Corollary (2.9)). These results are used to determine the structure of quasi-injective and quasi-projective modules over bounded (hnp)-rings (Theorems (2.13), (2.14) and (2.15)).

m-potent commutators involving skew derivations and multilinear polynomials

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Mohammad Ashraf ◽  
Sajad Ahmad Pary ◽  
Mohd Arif Raza
Keyword(s):  
Prime Ring ◽  
Quotient Ring ◽  
Multilinear Polynomial ◽  
Extended Centroid ◽  
Prime Rings ◽  
Skew Derivation ◽  
Martindale Quotient Ring ◽  
Multilinear Polynomials ◽  
The Right ◽  
The Relationship

AbstractLet {\mathscr{R}} be a prime ring, {\mathscr{Q}_{r}} the right Martindale quotient ring of {\mathscr{R}} and {\mathscr{C}} the extended centroid of {\mathscr{R}}. In this paper, we discuss the relationship between the structure of prime rings and the behavior of skew derivations on multilinear polynomials. More precisely, we investigate the m-potent commutators of skew derivations involving multilinear polynomials, i.e.,\big{(}[\delta(f(x_{1},\ldots,x_{n})),f(x_{1},\ldots,x_{n})]\big{)}^{m}=[% \delta(f(x_{1},\ldots,x_{n})),f(x_{1},\ldots,x_{n})],where {1<m\in\mathbb{Z}^{+}}, {f(x_{1},x_{2},\ldots,x_{n})} is a non-central multilinear polynomial over {\mathscr{C}} and δ is a skew derivation of {\mathscr{R}}.

scholarly journals σ- ideals and generalized derivations in σ-prime rings

2013 ◽  
Vol 31 (2) ◽  
pp. 113
Author(s):  
M. Rais Khan ◽  
Deepa Arora ◽  
M. Ali Khan
Keyword(s):  
Prime Ring ◽  
Generalized Derivations ◽  
Prime Rings

Let R be a prime ring and F and G be generalized derivations of R with associated derivations d and g respectively. In the present paper, we shall investigate the commutativity of R admitting generalized derivations F and G satisfying any one of the properties: (i) F(x)x = x G(x), (ii) F(x2) = x2 , (iii) [F(x), y] = [x, G(y)], (iv) d(x)F(y) = xy, (v) F([x, y]) = [F(x), y] + [d(y), x] and (vi) F(x ◦ y) = F(x) ◦ y − d(y) ◦ x for all x, y in some appropriate subset of R.

scholarly journals Generalized Left Jordan ideals In Prime Rings

2019 ◽  
Vol 17 (72) ◽  
pp. 87-92
Author(s):  
Kassim A. Jassim ◽  
Ali Kareem Kadhim
Keyword(s):  
Prime Ring ◽  
Prime Rings

     Let R be a prime ring and U be a (σ,τ)-left Jordan ideal .Then in this paper, we proved the following , if aU Z (Ua Z), a R, then a = 0 or U Z. If aU C s,t (Ua  C s,t), a R, then  either a = 0   or   U Z. If  0 ≠ [U,U] s,t .Then U Z. If  0≠[U,U] s,t C s,t, then   U Z  .Also, we checked the converse  some of these theorems and showed that are not true, so we give an example for them.

scholarly journals Multiplicative (generalized)-derivations of prime rings that act as $n$-(anti)homomorphisms

2020 ◽  
Vol 53 (2) ◽  
pp. 125-133
Author(s):  
G.S. Sandhu
Keyword(s):  
Prime Ring ◽  
Generalized Derivations ◽  
Prime Rings ◽  
The Given

Let R be a prime ring. In this note, we describe the possible forms of multiplicative (generalized)-derivations of R that act as n-homomorphism or n-antihomomorphism on nonzero ideals of R. Consequently, from the given results one can easily deduce the results of Gusić ([7]).

scholarly journals On Higher N-Derivation Of Prime Rings

2014 ◽  
Vol 11 (2) ◽  
pp. 211-219
Author(s):  
Baghdad Science Journal
Keyword(s):  
Prime Ring ◽  
Prime Rings ◽  
Torsion Free

The main purpose of this work is to introduce the concept of higher N-derivation and study this concept into 2-torsion free prime ring we proved that:Let R be a prime ring of char. 2, U be a Jordan ideal of R and be a higher N-derivation of R, then , for all u U , r R , n N .

scholarly journals Semi derivations of prime gamma rings

2012 ◽  
Vol 31 ◽  
pp. 65-70
Author(s):  
Kalyan Kumar Dey ◽  
Akhil Chandra Paul
Keyword(s):  
Prime Ring ◽  
Additive Mapping ◽  
Prime Rings ◽  
Subject Classifications ◽  
Associated Function ◽  
Gamma Rings

Let M be a prime ?-ring satisfying a certain assumption (*). An additive mapping f : M ? M is a semi-derivation if f(x?y) = f(x)?g(y) + x?f(y) = f(x)?y + g(x)?f(y) and f(g(x)) = g(f(x)) for all x, y?M and ? ? ?, where g : M?M is an associated function. In this paper, we generalize some properties of prime rings with semi-derivations to the prime &Gamma-rings with semi-derivations. 2000 AMS Subject Classifications: 16A70, 16A72, 16A10.DOI: http://dx.doi.org/10.3329/ganit.v31i0.10309GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 31 (2011) 65-70

Some results on ∗-differential identities in prime rings

2021 ◽  
Author(s):  
Deepak Kumar ◽  
Bharat Bhushan ◽  
Gurninder S. Sandhu
Keyword(s):  
Prime Ring ◽  
Generalized Derivation ◽  
Additive Mapping ◽  
Generalized Derivations ◽  
Prime Rings ◽  
Differential Identities ◽  
Derivation Formula

Let [Formula: see text] be a prime ring with involution ∗ of the second kind. An additive mapping [Formula: see text] is called generalized derivation if there exists a unique derivation [Formula: see text] such that [Formula: see text] for all [Formula: see text] In this paper, we investigate the structure of [Formula: see text] and describe the possible forms of generalized derivations of [Formula: see text] that satisfy specific ∗-differential identities. Precisely, we study the following situations: (i) [Formula: see text] (ii) [Formula: see text] (iii) [Formula: see text] (iv) [Formula: see text] for all [Formula: see text] Moreover, we construct some examples showing that the restrictions imposed in the hypotheses of our theorems are not redundant.

b-Generalized Derivations Acting on Multilinear Polynomials in Prime Rings

2018 ◽  
Vol 25 (04) ◽  
pp. 681-700
Author(s):  
Basudeb Dhara ◽  
Vincenzo De Filippis
Keyword(s):  
Prime Ring ◽  
Multilinear Polynomial ◽  
Extended Centroid ◽  
Generalized Derivations ◽  
Prime Rings ◽  
Ring Of Quotients ◽  
Multilinear Polynomials

Let R be a prime ring of characteristic different from 2, Q be its maximal right ring of quotients, and C be its extended centroid. Suppose that [Formula: see text] is a non-central multilinear polynomial over C, [Formula: see text], and F, G are two b-generalized derivations of R. In this paper we describe all possible forms of F and G in the case [Formula: see text] for all [Formula: see text] in Rn.

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