Notes on symmetric bi-(α,α)-derivations in rings Golbasi, O.; Sogutku, E.K.
Bulletin of the Transilvania University of Brasov. Series III: Mathematics and Computer Science,
2023, Letnik:
3, Številka:
1
Journal Article
Let R be a prime ring with center Z, I a nonzero ideal of R and D: R × R → R a symmetric bi–(α, α)-derivation and d be the trace of D. In the present paper, we have considered the following ...conditions: i) d(x), xα,α = 0, ii)d(x), xα,α ⊆ Cα,α, iii)(d(x), x)α,α = 0, iv)D1(d2(x), x) = 0, v)d1(d2(x)) = f(x), for all x, y ∈ I, where D1 and D2 are two symmetric bi-(α, α)-derivations, d1, d2 are the traces of D1, D2 respectively, B: R × R → R is a symmetric bi-additive mapping, f is the trace of B.
Let N be a left near-ring and S be a nonempty subset of N. A mapping F from N to N is called commuting on S if F(x),x = 0 for all x € S. The mapping F is called strong commutativity preserving (SCP) ...on S if F(x),F(y) = x,y for all x, y € S. In the present paper, firstly we generalize the well known result of Posner which is commuting derivations on prime rings to generalized derivations of semiprime near-rings. Secondly, we investigate SCP-generalized derivations of prime near-rings.
Let R be a semiprime ring and I a nonzero ideal of R. A map F:R→R is called a multiplicative generalized derivation if there exists a map d:R→R such that F(xy) = F(x)y+xd(y), for all x,y∈R. In the ...present paper, we shall prove that R contains a nonzero central ideal if any one of the following holds: i)
ii)
iii) F is SCP on I, iv) F(u)∘F(v) = u∘v, for all u,v∈I.
Let
R
will be a
2
-torsion free
∗
-prime ring and
α
,
β
∈
A
u
t
R
.
F
be a nonzero generalized
(
α
,
β
)
-derivation of
R
with associated nonzero
(
α
,
β
)
-derivation
d
which commutes with
∗
and
J
...be a nonzero
∗
-Jordan ideal and a subring of
R
.
In the present paper, we shall prove that
R
is commutative if any one of the following holds: (i)
F
(
u
)
,
u
α
,
β
=
0
,
(ii)
F
(
u
)
α
(
u
)
=
β
(
u
)
d
(
u
)
,
(iii)
F
(
u
2
)
=
±
α
(
u
2
)
,
(iv)
F
(
u
2
)
=
2
d
(
u
)
α
(
u
)
,
(v)
d
(
u
2
)
=
2
F
(
u
)
α
(
u
)
,
for all
u
∈
U
.
In the present paper, we shall prove that 3-prime near-ring N is commutative ring, if any one of the following conditions are satisfied: (i) f(N)⊆Z, (ii) f(x,y)=0, (iii) f(x,y)=±x,y, (iv) ...f(x,y)=±(xoy), (v) f(x,y)=f(x),y, (vi) f(x,y)=x,f(y), (vii) f(x,y)=d(x),y, (viii) f(x,y)=d(x)oy,(ix) f(x),y∈Z for all x,y∈N where f is a nonzero multiplicative generalized derivation of N associated with a multiplicative derivation d.
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