Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def ...\Rey {\mathit {Re}}R$ be a semiprime ring with extended centroid $C$ and with maximal right ring of quotients $Q_{mr}(R)$. Let $d{:}\ R\to Q_{mr}(R)$ be an additive map and $b\in Q_{mr}(R)$. An additive map $\delta {:}\ R\to Q_{mr}(R)$ is called a (left) $b$-generalized derivation with associated map $d$ if $\delta (xy)=\delta (x)y+bxd(y)$ for all $x, y\in R$. This gives a unified viewpoint of derivations, generalized derivations and generalized $\sigma $-derivations with an X-inner automorphism $\sigma $. We give a complete characterization of $b$-generalized derivations of $R$ having nilpotent values of bounded index. This extends several known results in the literature.
A classical result of Herstein asserts that any Jordan derivation on a prime ring of characteristic different from two is a derivation. It is our aim in this paper to prove the following result, ...which is in the spirit of Herstein's theorem. Let \(n≥ 3\) be some fixed integer, let R be a prime ring with \(char(R)> 4n-8\) and let D:R → R be an additive mapping satisfying either the relation \(D(x^n)=D(x^{n-1})x+x^{n-1}D(x)\) or the relation \(D(x^n)=D(x)x^{n-1}+xD(x^{n-1})\) for all \(x \in R\). In both cases D is a derivation.
Let R be a ring (not necessarily with 1) and G be a finite group of automorphisms of R. The set
of primes p such that
and R is not p-torsion free, is called the set of bad primes. When the ring is
...-torsion free, i.e.
the properties of the rings R and R
G
are closely connected. The aim of the article is to show that this is also true when
under natural conditions on bad primes. In particular, it is shown that the Jacobson radical (respectively, the prime radical) of the ring R
G
is equal to the intersection of the Jacobson radical (respectively, the prime radical) of R with R
G
; if the ring R is semiprime then so is R
G
; if the trace of the ring R is nilpotent then the ring itself is nilpotent; if R is a semiprime ring then R is left Goldie iff the ring R
G
is so, and in this case, the ring of G-invariants of the left quotient ring of R is isomorphic to the left quotient ring of R
G
and
The purpose of this paper is to prove some results concerning permuting triderivations and permuting generalized triderivations on prime and semiprime rings which partially extend some results ...contained in 9 and 10.
In this paper, we give an affirmative answer to two conjectures on generalized (
m
,
n
)-Jordan derivations and generalized (
m
,
n
)-Jordan centralizers raised in Ali and Fošner (Algebra Colloq ...21:411–420, 2014) and Fošner (Demonstr Math 46:254–262, 2013). Precisely, when
R
is a semiprime ring, we prove, under some suitable torsion restrictions, that every nonzero generalized (
m
,
n
)-Jordan derivation (resp., a generalized (
m
,
n
)-Jordan centralizer) is a derivation (resp., a two-sided centralizer).
Centrally essential rings were defined earlier for associative unital rings; in this paper, we define them for rings which are not necessarily associative or unital. In this case, it is proved that ...centrally essential semiprime rings are commutative. It is proved that all idempotents of a centrally essential alternative ring are central. Several examples of non-commutative centrally essential rings are provided, some properties of centrally essential rings are described.
Let ℛ be a prime ring of characteristic different from 2, its right Ututmi quotient ring, its extended centroid, f(x
1
, ... , x
n
) a multilinear polynomial in n non-commuting variables over and S = ...{ f(r
1
, ... , r
n
) : r
1
, ... , r
n
∈ ℛ}. Let F: ℛ → ℛ and G: ℛ → ℛ be non-zero generalized derivations on ℛ. We say that F and G are mutually strong Engel condition preserving (SEP for brevity) on if G(x), F(y)
h
= x, y
h
, for all x, y ∈ and fixed h ≥ 1. In this article we show that, if f(x
1
, ... , x
n
) is not central valued on ℛ and F, G are mutually SEP on , then one of the following holds:
(a) there exists λ ∈ such that, for any x ∈ ℛ, G(x) = λx and F(x) = λ
−h
x;
(b)
char(R) = p ≥ 3 and there exist λ ∈ and s ≥ 1 such that, for any x ∈ ℛ, G(x) = λx and
is central valued on ℛ;
(c) ℛ satisfies s
4
, the standard identity of degree 4.
The semiprime case for mutually SEP derivations on Lie ideals is also considered.
Prime structures in a Morita context Calci, Mete Burak; Halicioglu, Sait; Harmanci, Abdullah ...
Boletín de la Sociedad Matemática Mexicana,
11/2020, Volume:
26, Issue:
3
Journal Article
Peer reviewed
Open access
In this paper, we study the primeness and semiprimeness of a Morita context. Necessary and sufficient conditions are investigated for an ideal of a Morita context to be a prime ideal and a semiprime ...ideal. In particular, we determine the conditions under which a Morita context is prime and semiprime.
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