In the present paper, we prove some commutativity theorems for a prime ring with involution in which generalized derivations satisfy certain differential identities. Some well known results on ...commutativity of prime rings have been obtained. Also, we provide an example to show that the assumed restriction imposed on the involution is not superfluous.
On Derivations in Semiprime Rings Ali, Shakir; Shuliang, Huang
Algebras and representation theory,
12/2012, Letnik:
15, Številka:
6
Journal Article
Recenzirano
Let
R
be a ring,
S
a nonempty subset of
R
and
d
a derivation on
R
. A mapping
is called commuting on
S
if
f
(
x
),
x
= 0 for all
x
∈
S
. In this paper, our purpose is to produce commutativity ...results for rings and show that if
R
is a 2-torsion free semiprime ring and
I
a nonzero ideal of
R
, then a derivation
d
of
R
is commuting on
I
if one of the following conditions holds: (i)
d
(
x
) ∘
d
(
y
) =
x
∘
y
(ii)
d
(
x
) ∘
d
(
y
) = − (
x
∘
y
) (iii)
d
(
x
) ∘
d
(
y
) = 0 (iv)
d
(
x
),
d
(
y
) = −
x
,
y
(v)
d
(
x
)
d
(
y
) =
xy
(vi)
d
(
x
)
d
(
y
) = −
xy
(vii)
d
(
x
)
d
(
y
) =
yx
(viii)
d
(
x
)
d
(
x
) =
x
2
for all
x
,
y
∈
I
. Further, if
d
(
I
) ≠ 0, then
R
has a nonzero central ideal. Finally, some examples are given to demonstrate that the restrictions imposed on the hypotheses of the various results are not superfluous.
NHR-200-II is a small integrated pressurized water reactor with 200 MW core thermal power. The core heat is transferred to two independent intermediate circuits via fourteen in-vessel primary heat ...exchangers (PHE), and the heat in the intermediate circuits is transferred to feedwater by two steam generators (SG) in the two intermediate circuits respectively. A passive residual heat removal (PRHR) branch is connected to each intermediate circuit to remove core decay heat under postulated accidents. During normal operation, PRHR branches are isolated by valves while SG branches in intermediate circuits are open. The valves in PRHR branches will be opened and the isolation valves of SG branches will be closed during decay heat removal scenarios. The decay heat removal capacity of NHR-200-II PRHRS could be seriously deteriorated once the isolation valves for SG branches fail to close, which was confirmed in a scaled integral test loop previously. Current understanding of PRHRS’s thermal-hydraulic characteristics with possible isolation failure in SG branches is limited. In this paper, the NHR-200-II PRHRS is modeled with RELAP5 considering the case of success and fail to isolate SG branches. A series of numerical simulations are carried out to study the impact of various parameters, such as the initial temperature, the size of the intermediate circuits’ header, and the initial flow direction in the intermediate circuits. Oscillatory flow is found when SG branches fail to be isolated under certain parameters combinations. An improved PRHRS design is purposed to eliminate possible flow oscillations, and the purposed improved design are tested by numerical simulations.
Let R be a 2-torsion-free prime ring with center Z(R), F a generalized derivation associated with a nonzero derivation d, L a Lie ideal of R. If (d(u)l1F(u)l2d(u)l3F(u)l4…F(u)lk)n=0 for all u∈L, ...where l1,l2,…,lk are fixed non-negative integers not all zero, and n is fixed positive integer, then L⊆Z(R). We also examine the case when R is a semiprime ring. Finally, we apply the above result to Banach algebras.
Let R be a 2-torsion free sigma-prime ring with an involution sigma, I a nonzero sigma-ideal of R. In this paper we explore the commutativity of R satisfying any one of the properties: (i)d(x)◦F(y) = ...0 for all x, y ∈ I. (ii) d(x),F(y) = 0 for all x,y ∈ I. (iii) d(x)◦F(y) = x◦y for all x, y ∈ I. (iv) d(x)F(y) − xy ∈ Z(R) for all x, y ∈ I. We also discuss (alpha,beta)-derivations of sigma-prime rings and prove that if G is an (alpha,beta)-derivation which acts as a homomorphism or as an anti-homomorphism on I, then G = 0 or G = on I.
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.
In this paper, we present some results concerning orthogonal generalized (σ,τ)-derivations in semiprime near-rings. These results are a generalization of results of Bresar and Vukman, which are ...related to a theorem of Posner for the product of two derivations in prime rings. KCI Citation Count: 0
On derivations of prime and semi-prime Gamma rings Kamali Ardakani, Leili; Davvaz, Bijan; Huang, Shuliang
Boletim da Sociedade Paranaense de Matemática,
01/2019, Letnik:
37, Številka:
2
Journal Article
Recenzirano
Odprti dostop
The concept of $\Gamma$-ring is a generalization of ring. Twoimportant classes of $\Gamma$-rings are prime and semi-prime$\Gamma$-rings. In this paper, we consider the concept of derivations on prime ...and semi-prime $\Gamma$-rings and we study some of their properties.
Generalized Derivations of Prime Rings Shuliang, Huang
International Journal of Mathematics and Mathematical Sciences,
2007, Letnik:
2007
Journal Article
Recenzirano
Odprti dostop
Let
R
be an associative prime ring,
U
a Lie ideal such that
u
2
∈
U
for all
u
∈
U
. An additive function
F
:
R
→
R
is called a generalized derivation if there exists a derivation
d
:
R
→
R
such that
...F
(
x
y
)
=
F
(
x
)
y
+
x
d
(
y
)
holds for all
x
,
y
∈
R
. In this paper, we prove that
d
=
0
or
U
⊆
Z
(
R
)
if any one of the following conditions holds: (1)
d
(
x
)
∘
F
(
y
)
=
0
, (2)
d
(
x
)
,
F
(
y
)
=
0
, (3) either
d
(
x
)
∘
F
(
y
)
=
x
∘
y
or
d
(
x
)
∘
F
(
y
)
+
x
∘
y
=
0
, (4) either
d
(
x
)
∘
F
(
y
)
=
x
,
y
or
d
(
x
)
∘
F
(
y
)
+
x
,
y
=
0
, (5) either
d
(
x
)
∘
F
(
y
)
−
x
y
∈
Z
(
R
)
or
d
(
x
)
∘
F
(
y
)
+
x
y
∈
Z
(
R
)
, (6) either
d
(
x
)
,
F
(
y
)
=
x
,
y
or
d
(
x
)
,
F
(
y
)
+
x
,
y
=
0
, (7) either
d
(
x
)
,
F
(
y
)
=
x
∘
y
or
d
(
x
)
,
F
(
y
)
+
x
∘
y
=
0
for all
x
,
y
∈
U
.
Let
be a prime ring with center
) and
a nonzero right ideal of
. Suppose that
admits a generalized reverse derivation (
,
) such that
)) ≠ 0. In the present paper, we shall prove that if one of the ...following conditions holds:
(i)
(
) ±
∈
(ii)
(
,
) ±
(
),
∈
(iii)
(
)
(
),
(
) ∈
(iv)
(
ο
) ±
(
) ο
(
) ∈
(v)
(
),
±
,
(
) ∈
(vi)
(
) ο
ο
(
) ∈
for all
∈
, then
is commutative.
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