J.
Indones.
Math.
Soc.
Vol.
No.
, pp.
44Ae51.
SOME IDENTITIES INVOLVING MULTIPLICATIVE
(GENERALIZED) (, .
-DERIVATIONS IN SEMIPRIME
RINGS
Naga Malleswari1 .
Sreenivasulu2 , and G.
Shobhalatha1 Department of Mathematics.
Sri Krishnadevaraya University.
Anantapur-515003, malleswari.
gn@gmail.
Department of Mathematics.
Government College (Autonomou.
Anantapur-515001 Abstract.
Let R be a semiprime ring.
I a nonzero ideal of R and be an automorphism of R.
A map F : R OeIe R is said to be a multiplicative .
(, .
-derivation associated with a map d : R OeIe R such that F .
= F .
, for all x, y OO R.
In the present paper, we shall prove that R contains a nonzero central ideal if any one of the following holds: .
F .
, .
A .
, .
= 0, .
F .
A .
= 0, .
F .
, .
= [F .
, .
,1 , .
F .
, .
= (F .
,1 , .
F .
= [F .
, .
,1 and .
F .
= (F .
,1 , for all x, y OO I.
Key words and Phrases:
Ideal.
Semiprime rings.
Multiplicative .
(, .
- INTRODUCTION Let R be an associative ring with center Z.
For any x, y OO R, the symbol .
, .
stands for the commutator xy Oe yx and symbol x y denotes for the anticommutator xy yx.
Recall, a ring R is prime ring if xRy = 0 implies x = 0 or y = 0 and R is semiprime ring if xRx = 0 implies x = 0.
Let and be automorphisms of R.
For any x, y OO R, .
, .
, = x.
Oe .
x and .
, = x.
By considering = 1, where 1 is an identity mapping on R, we have .
, .
,1 = x.
Oe yx and .
,1 = x.
An additive mapping d : R OeIe R is called a derivation if d.
= d.
y xd.
holds for all x, y OO R.
The concept of a derivation was extended to generalized derivation by Bresar .
An additive mapping F : R OeIe R is said to be a generalized derivation if there exists a derivation d : R OeIe R such that F .
= F .
y xd.
for all x, y OO R.
2020 Mathematics Subject Classification: 16N60, 16W25.
Received: 18-03-2021, accepted: 28-10-2021.
Some Identities Involving Multiplicative(Generalize.
(, .
-Derivations Inspired by the work of Martindale i .
Daif .
introduced the concept of multiplicative derivations.
Accordingly, a map d : R OeIe R is called a multiplicative derivation of R if d.
= d.
y xd.
holds for all x, y OO R.
Of course, these maps are not necessarily additive.
Then the complete description of these maps was given by Goldman and Semrl .
Further.
Daif and Tammam-El-Sayiad .
extended the notion of multiplicative derivation to multiplicative generalized derivation of R if F .
= F .
y xd.
holds for all x, y OO R, where d is derivation on Recently, the definition of multiplicative generalized derivation was extended to multiplicative .
-derivation by Dhara and Ali .
as follows: a map F : R OeIe R .
ot necessarily additiv.
is said to be a multiplicative .
derivation if F .
= F .
y xd.
holds for all x, y OO R, where d can be any map.
ot necessarily additive nor a derivatio.
Chang .
introduced the notion of a generalized(, )-derivation of a ring R and investigated some properties of such derivations.
let , be mappings of R into itself.
An additive mapping F : R OeIe R is called a generalized (, )derivation of R such that F .
= F .
for all x, y OO R where and are automorphisms on R.
A mapping F : R OeIe R is said to be a multiplicative .
(, )-derivation if there exists a map d on R such that F .
= F .
for all x, y OO R.
Obviously every generalized (, )-derivation is a multiplicative .
(, )-derivation.
In 1992.
Daif .
, proved a result that if R is a semiprime ring.
I be a non-zero ideal of R and d is a derivation of R such that d (.
, .
) = A .
, .
for all x, y OO I, then I OI Z(R).
Quadri .
extended the result of Daif by replacing derivation d with a generalized derivation in a prime ring.
Recently, shauliang .
studied the identities related to generalized (, ) derivation on prime rings.
Asma Ali et al.
studied the identities related to multiplicative .
(, )-derivations in semiprime In this line of investigation, in the present paper we shall prove that R contains a non-zero central ideal if any one of the following holds: .
F .
, .
A .
, .
= 0, .
F .
A .
= 0, .
F .
, .
= [F .
, .
,1 , .
F .
, .
= (F .
,1 , .
F .
= [F .
, .
,1 , .
F .
= (F .
,1 , for all x, y OO Throughout the present paper, we shall make use of the following basic identities without any specific mention:
, y.
= y .
, .
, .
z, .
y, .
= .
, .
y x .
, .
, .
x yz = .
z Oe y .
, .
= y .
, .
z, .
xy z = x .
Oe .
, .
y = .
y x .
, .
, .
y, .
,1 = x .
, .
,1 .
, .
y = x .
, .
] .
, .
,1 y, .
, y.
,1 = y .
, .
,1 .
, .
,1 .
, .
),1 = .
,1 .
Oe y .
, .
,1 = y .
,1 .
, .
,1 .
, .
,1 = x .
,1 Oe .
, .
y = .
,1 y x .
, .
] .
Malleswari.
Sreenivasulu, and Shobhalatha
MAIN RESULTS
In order to prove our main theorems, we shall need the following lemma.
Lemma 2.
Lemma 2.
) Let R be a semiprime ring and I is a nonzero two sided ideal of R and a OO R such that axa = 0 for all x OO I, then a = 0.
Theorem 2.
Let R be a semiprime ring.
I a nonzero ideal of R and is an automorphism of R.
Suppose that F is multiplicative .
(, .
-derivation on R associated with the map d on R.
If F .
, .
A .
, .
= 0 holds for all x, y OO I, then d is commuting on I.
Proof.
By the hypothesis, we have F .
, .
A .
, .
= 0 for all x, y OO I.
Replacing y by yx in .
, we obtain that F (.
, .
A (.
, .
= 0 for all x, y OO I, and so F (.
, .
) .
, .
A (.
, .
) .
= 0 for all x, y OO I.
Using the hypothesis, we obtain .
, .
= 0 for all x, y OO I.
Replacing y by ry in .
, we get r .
, .
, .
= 0 for all x, y OO I, r OO R.
Using .
, we obtain .
, .
= 0 for all x, y OO I, r OO R.
Replacing y by yx in .
, we get .
, .
= 0 for all x, y OO I, r OO R.
Right multiplying .
by x, we have .
, .
x = 0 for all x, y OO I, r OO R.
Subtracting .
, we get .
, .
, d .
] = 0 for all x, y OO I, r OO R.
Replacing r by d .
in the last equation, we have .
, d .
] y .
, d .
] = 0 for all x, y OO I.
That is .
, d .
] I .
, d .
] = 0 for all x OO I.
By lemma 2.
1, we conclude that .
, d .
] = 0 for all x OO I.
Therefore d is commuting on I.
Some Identities Involving Multiplicative(Generalize.
(, .
-Derivations Theorem 2.
Let R be a semiprime ring.
I a nonzero ideal of R and is an automorphism of R.
Suppose that F is multiplicative .
(, .
-derivation on R associated with the map d.
If F .
A .
= 0 holds for all x, y OO I, then d is commuting on I.
Proof.
By the hypothesis, we have F .
A .
= 0 for all x, y OO I.
Replacing y by yx in .
, we obtain that F (.
A (.
= 0 for all x, y OO I, and so F (.
) .
A (.
) .
= 0 for all x, y OO I.
Using the hypothesis, we obtain .
= 0 for all x, y OO I.
Replacing y by ry in .
, we find that r .
, .
= 0 for all x, y OO I, r OO R.
Using .
, we obtain .
, .
= 0 for all x, y OO I, r OO R.
Using the same arguments as used in the proof of Theorem 2.
2, we get the required Theorem 2.
Let R be a semiprime ring.
I a nonzero ideal of R and is an automorphism of R.
Suppose that F is multiplicative .
(, .
-derivation on R associated with the map d.
If F .
, .
= [F .
, .
,1 holds for all x, y OO I, then d is commuting on I.
Proof.
By the hypothesis, we have F .
, .
= [F .
, .
,1 for all x, y OO I.
Replacing y by yx in .
, we obtain that F (.
, .
= y [F .
, .
,1 [F .
, .
,1 .
for all x, y OO I, and so F (.
, .
) .
, .
= y [F .
, .
,1 [F .
, .
,1 .
for all x, y OO I.
Using the hypothesis, we obtain .
, .
= y [F .
, .
,1 for all x, y OO I.
Replacing y by ry in .
, we find that r .
, .
, .
= ry [F .
, .
,1 for all x, y OO I, r OO R.
Using .
, we get .
, .
= 0 for all x, y OO I, r OO R.
Malleswari.
Sreenivasulu, and Shobhalatha Using similar argument as used in the proof of Theorem 2.
2, we get the required Theorem 2.
Let R be a semiprime ring.
I a nonzero ideal of R and is an automorphism of R.
Suppose that F is multiplicative .
(, .
-derivation on R associated with the map d.
If F .
, .
= (F .
,1 holds for all x, y OO I, then d is commuting on I.
Proof.
By the hypothesis, we have F .
, .
= (F .
,1 for all x, y OO I.
Replacing y by yx in .
, we obtain that F (.
, .
= (F .
,1 .
Oe y [F .
, .
,1 for all x, y OO I and so.
F (.
, .
) .
, .
= (F .
,1 .
Oe y [F .
, .
,1 for all x, y OO I.
Using the hypothesis, we obtain .
, .
= Oey [F .
, .
,1 for all x, y OO I.
Replacing y by ry in .
, we get r .
, .
, .
= Oery [F .
, .
,1 for all x, y OO I, r OO R.
Using .
, we get .
, .
= 0 for all x, y OO I, r OO R.
Arguing in the similar manner as in Theorem 2.
2, we get the result.
Theorem 2.
Let R be a semiprime ring.
I a nonzero ideal of R and is an automorphism of R.
Suppose that F is multiplicative .
(, .
-derivation on R associated with the map d.
If F .
= [F .
, .
,1 holds for all x, y OO I, then d is commuting on I.
Proof.
By the hypothesis, we have F .
= [F .
, .
,1 for all x, y OO I.
Replacing y by yx in .
, we obtain that F (.
= y [F .
, .
,1 [F .
, .
,1 .
for all x, y OO I, and so F (.
) .
= y [F .
, .
,1 [F .
, .
,1 .
for all x, y OO I.
Using the hypothesis, we obtain .
= y [F .
, .
,1 for all x, y OO I.
Replacing y by ry in .
, we get r .
, .
= ry [F .
, .
,1 for all x, y OO I, r OO R.
Some Identities Involving Multiplicative(Generalize.
(, .
-Derivations Using .
, we have .
, .
= 0 for all x, y OO I, r OO R.
Arguing in the similar manner as in Theorem 2.
2, we get the result.
Theorem 2.
Let R be a semiprime ring.
I a nonzero ideal of R and is an automorphism of R.
Suppose that F is multiplicative .
(, .
-derivation on R associated with the map d.
If F .
= (F .
,1 holds for all x, y OO I, then d is commuting on I.
Proof.
By the hypothesis, we have F .
= (F .
,1 for all x, y OO I.
Replacing y by yx in .
, we obtain that F (.
= (F .
,1 .
Oe y [F .
, .
,1 for all x, y OO I, and so F (.
) .
= (F .
,1 .
Oe y [F .
, .
,1 for all x, y OO I.
Using the hypothesis, we obtain .
= Oey [F .
, .
,1 for all x, y OO I.
Replacing y by ry in .
, we find that r .
, .
= Oery [F .
, .
,1 for all x, y OO I, r OO R.
Using .
, we have .
, .
= 0 for all x, y OO I, r OO R.
Arguing in the similar manner as in Theorem 2.
2, we get the result.
Corollary 2.
Let R be a semiprime ring.
Suppose that F, d is a multiplicative .
(, .
-derivation of R.
If any one of the following holds:
F .
, .
A .
, .
= 0 .
F .
A .
= 0 .
F .
, .
= [F .
, .
,1 .
F .
, .
= (F .
,1 .
F .
= [F .
, .
,1 .
F .
= (F .
,1 OAx, y OO R then d is commuting on R.
Malleswari.
Sreenivasulu, and Shobhalatha Example In this, we construct an example to the condition .
of corollary 2.
8 so that the semiprimeness condition of the ring is essential.
Example 1.
Let Z be the set of integers and R = | a, b, c OO Z .
I | a, b, c OO Z .
Let us define F, d, : R OeIe R by F 0 Oeb 0 Oeb a Oeb It is easy to verify that I is an ideal on R.
F is multiplicative .
(, .
-derivation associated with the map d, is an automorphism on R.
We see that , but is nonzero element of R.
It implies that R is not semiprime.
Acknowledgement.
The authors are very thankful to the referees for his/her careful reading of the paper and valuable comments.
REFERENCES