Modules with few types over a hereditary noetherian prime ring

2001 ◽  
Vol 66 (1) ◽  
pp. 271-280 ◽  
Author(s):  
Vera Puninskaya
Keyword(s):  
Prime Ring ◽  
Hereditary Noetherian Prime Ring

AbstractIt is proved that Vaught's conjecture is true for modules over an arbitrary countable hereditary noetherian prime ring.

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)).

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.

A Note on Differential Identities in Prime and Semiprime Rings

2020 ◽  
pp. 77-83
Author(s):  
Mohammad Shadab Khan ◽  
Mohd Arif Raza ◽  
Nadeemur Rehman
Keyword(s):  
Prime Ring ◽  
Semiprime Ring ◽  
Nonzero Ideal ◽  
Differential Identities ◽  
Standard Identity ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings ◽  
Positive Integers

Let R be a prime ring, I a nonzero ideal of R, d a derivation of R and m, n fixed positive integers. (i) If (d ( r ○ s)(r ○ s) + ( r ○ s) d ( r ○ s)n - d ( r ○ s))m for all r, s ϵ I, then R is commutative. (ii) If (d ( r ○ s)( r ○ s) + ( r ○ s) d ( r ○ s)n - d (r ○ s))m ϵ Z(R) for all r, s ϵ I, then R satisfies s4, the standard identity in four variables. Moreover, we also examine the case when R is a semiprime ring.

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 derivations in prime and semiprime

2015 ◽  
Vol 34 (2) ◽  
pp. 29
Author(s):  
Shuliang Huang ◽  
Nadeem Ur Rehman
Keyword(s):  
Prime Ring ◽  
Semiprime Ring ◽  
Generalized Derivation ◽  
Nonzero Ideal ◽  
Generalized Derivations ◽  
Positive Integers

Let $R$ be a prime ring, $I$ a nonzero ideal of $R$ and $m, n$  fixed positive integers.  If $R$ admits a generalized derivation $F$ associated with a  nonzero derivation $d$ such that $(F([x,y])^{m}=[x,y]_{n}$ for  all $x,y\in I$, then $R$ is commutative. Moreover  we also examine the case when $R$ is a semiprime ring.

scholarly journals Identities with derivations and automorphisms on semiprime rings

2005 ◽  
Vol 2005 (7) ◽  
pp. 1031-1038 ◽  
Author(s):  
Joso Vukman
Keyword(s):  
Prime Ring ◽  
Classical Result ◽  
Semiprime Rings

The purpose of this paper is to investigate identities with derivations and automorphisms on semiprime rings. A classical result of Posner states that the existence of a nonzero centralizing derivation on a prime ring forces the ring to be commutative. Mayne proved that in case there exists a nontrivial centralizing automorphism on a prime ring, then the ring is commutative. In this paper, some results related to Posner's theorem as well as to Mayne's theorem are proved.

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]).

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