On $\alpha$-Short Modules

2014 ◽  
Vol 114 (1) ◽  
pp. 26 ◽  
Author(s):  
M. Davoudian ◽  
O. A. S. Karamzadeh ◽  
N. Shirali
Keyword(s):  
Semiprime Ring ◽  
Countable Ordinal ◽  
Noetherian Dimension

We introduce and study the concept of $\alpha$-short modules (a $0$-short module is just a short module, i.e., for each submodule $N$ of a module $M$, either $N$ or $\frac{M}{N}$ is Noetherian). Using this concept we extend some of the basic results of short modules to $\alpha$-short modules. In particular, we show that if $M$ is an $\alpha$-short module, where $\alpha$ is a countable ordinal, then every submodule of $M$ is countably generated. We observe that if $M$ is an $\alpha$-short module then the Noetherian dimension of $M$ is either $\alpha$ or $\alpha+1$. In particular, if $R$ is a semiprime ring, then $R$ is $\alpha$-short as an $R$-module if and only if its Noetherian dimension is $\alpha$.

Dimension on non-essential submodules

2019 ◽  
Vol 18 (05) ◽  
pp. 1950089 ◽  
Author(s):  
Maryam Davoudian
Keyword(s):  
Associative Ring ◽  
Semiprime Ring ◽  
Krull Dimension ◽  
Goldie Dimension ◽  
Ordinal Numbers ◽  
Noetherian Dimension

In this paper, we introduce and study the concepts of non-essential Krull dimension and non-essential Noetherian dimension of an [Formula: see text]-module, where [Formula: see text] is an arbitrary associative ring. These dimensions are ordinal numbers and extend the notion of Krull dimension. They respectively rely on the behavior of descending and ascending chains of non-essential submodules. It is proved that each module with non-essential Krull dimension (respectively, non-essential Noetherian dimension) has finite Goldie dimension. We also show that a semiprime ring [Formula: see text] with non-essential Noetherian dimension is uniform.

On Traces of n-Additive Mappings on Semiprime Ring

2017 ◽  
Vol 4 (1) ◽  
pp. 1-10
Author(s):  
Najat Muthana ◽  
◽  
Asma Ali ◽  
Kapil Kumar
Keyword(s):  
Semiprime Ring ◽  
Additive Mappings

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.

Derivations of differentially semiprime rings

2019 ◽  
Vol 12 (05) ◽  
pp. 1950079
Author(s):  
Ahmad Al Khalaf ◽  
Iman Taha ◽  
Orest D. Artemovych ◽  
Abdullah Aljouiiee
Keyword(s):  
Commutative Ring ◽  
Semiprime Ring ◽  
Prime Rings ◽  
Commutative Case ◽  
Lie Ring ◽  
Lie Rings ◽  
Semiprime Rings ◽  
Torsion Free

Earlier D. A. Jordan, C. R. Jordan and D. S. Passman have investigated the properties of Lie rings Der [Formula: see text] of derivations in a commutative differentially prime rings [Formula: see text]. We study Lie rings Der [Formula: see text] in the non-commutative case and prove that if [Formula: see text] is a [Formula: see text]-torsion-free [Formula: see text]-semiprime ring, then [Formula: see text] is a semiprime Lie ring or [Formula: see text] is a commutative ring.

Two results on borel orders

1989 ◽  
Vol 54 (3) ◽  
pp. 865-874 ◽  
Author(s):  
Alain Louveau
Keyword(s):  
Linear Order ◽  
Linear Orders ◽  
Lexicographic Ordering ◽  
Countable Ordinal

AbstractWe prove two results about the embeddability relation between Borel linear orders: For η a countable ordinal, let 2η (resp. 2< η) be the set of sequences of zeros and ones of length η (resp. < η), equipped with the lexicographic ordering. Given a Borel linear order X and a countable ordinal ξ, we prove the following two facts.(a) Either X can be embedded (in a (X, ξ) way) in 2ωξ or 2ωξ + 1 continuously embeds in X.(b) Either X can embedded (in a (X, ξ) way) in 2<ωξ or 2ωξ continuously embeds in X. These results extend previous work of Harrington, Shelah and Marker.

A NOTE ON -JORDAN DERIVATIONS OF RINGS AND BANACH ALGEBRAS

2015 ◽  
Vol 93 (2) ◽  
pp. 231-237 ◽  
Author(s):  
IRENA KOSI-ULBL ◽  
JOSO VUKMAN
Keyword(s):  
Semiprime Ring ◽  
Banach Algebras ◽  
Additive Mapping ◽  
Jordan Derivation ◽  
Torsion Free

In this paper we prove the following result: let$m,n\geq 1$be distinct integers, let$R$be an$mn(m+n)|m-n|$-torsion free semiprime ring and let$D:R\rightarrow R$be an$(m,n)$-Jordan derivation, that is an additive mapping satisfying the relation$(m+n)D(x^{2})=2mD(x)x+2nxD(x)$for$x\in R$. Then$D$is a derivation which maps$R$into its centre.

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 Multiplicative (Generalized) Reverse Derivations on Semiprime Ring

2018 ◽  
Vol 11 (3) ◽  
pp. 717-729
Author(s):  
Asma Ali ◽  
Ambreen Bano
Keyword(s):  
Semiprime Ring ◽  
Additive Map

Let R be a semiprime ring. A mapping F : R → R (not necessarily additive) is called a multiplicative (generalized) reverse derivation if there exists a map    d : R → R (not necessarily a derivation nor an additive map) such that F(xy) = F(y)x + yd(x) for all x, y є R. In this paper we investigate some identities involving multiplicative (generalized) reverse derivation and prove some theorems in which we characterize these mappings.

Commutativity Conditions on Rings with Involution

1982 ◽  
Vol 34 (1) ◽  
pp. 17-22
Author(s):  
Paola Misso
Keyword(s):  
Semiprime Ring ◽  
Central Elements ◽  
Rings With Involution ◽  
Image Position ◽  
Partial Extension ◽  
Torsion Free

Let R be a ring with involution *. We denote by S, K and Z = Z(R) the symmetric, the skew and the central elements of R respectively.In [4] Herstein defined the hypercenter T(R) of a ring R asand he proved that in case R is without non-zero nil ideals then T(R) = Z(R).In this paper we offer a partial extension of this result to rings with involution.We focus our attention on the following subring of R:(We shall write H(R) as H whenever there is no confusion as to the ring in question.)Clearly H contains the central elements of R. Our aim is to show that in a semiprime ring R with involution which is 2 and 3-torsion free, the symmetric elements of H are central.

Additive n-commuting maps on semiprime rings

2019 ◽  
Vol 63 (1) ◽  
pp. 193-216
Author(s):  
Cheng-Kai Liu
Keyword(s):  
Unsolved Problem ◽  
Semiprime Ring ◽  
Affirmative Answer ◽  
Extended Centroid ◽  
Additive Map ◽  
Semiprime Rings ◽  
Ring Of Quotients ◽  
Functional Identities ◽  
Fixed Positive Integer ◽  
Commuting Maps

AbstractLet R be a semiprime ring with the extended centroid C and Q the maximal right ring of quotients of R. Set [y, x]1 = [y, x] = yx − xy for x, y ∈ Q and inductively [y, x]k = [[y, x]k−1, x] for k > 1. Suppose that f : R → Q is an additive map satisfying [f(x), x]n = 0 for all x ∈ R, where n is a fixed positive integer. Then it can be shown that there exist λ ∈ C and an additive map μ : R → C such that f(x) = λx + μ(x) for all x ∈ R. This gives the affirmative answer to the unsolved problem of such functional identities initiated by Brešar in 1996.

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