Commutative with b-Generalized Derivations in Prime Rings of Dense Ideal

 

Commutative with b-Generalized Derivations in Prime Rings of Dense Ideal

Dr K. Subbarayudu

Lecturer in Mathematics Government Polytechnic, Simhadripuram, AndhraPradesh, India

N. Subbarayudu

Research Scholar, Department of Mathematics, S.V.University Tirupati, Andhra Pradesh, India.

Dr A.Sivakameshwara Kumar

Department of Mathematics, CMR College of Engineering and Technology, Hyderabad, Telangana, India.

Dr C. Jaya Subba Reddy

Professor, Department of Mathematics, S.V.University Tirupati, Andhra Pradesh, India.

Abstract – Let be a prime ring, be a nonzero the dense ideal of and : be a b-generalized derivation associated with derivation : if (i) [, ] = 0 (ii) [, ] () = 0 (iii) [, ] [, ] = 0 then is commutative.

Keywords: Prime ring, Derivation, derivation, Generalized derivation, Generalized derivation, b-generalized derivations.

Introduction: The study of derivations in rings has evolved through various generalizations, including generalized derivations, multiplicative generalized derivations, skew generalized derivations, and b-generalized derivations. In 2014, the concept of the b-generalized derivation was first introduced by Koan and Lee [5]. In 2017 2018, research shifted toward Engel conditions. C.K. Liu [6] proved several results regarding Engel conditions with b-generalized derivations, which were later extended to Lie ideals by Liau et al. [7] in 2018. Simultaneously, De Filippis et al. [3] investigated b-generalized skew derivations on Lie ideals. In 2023, Mozumder et al. [9, 10] contributed a note on b-generalized derivations in rings with involution and extended these findings to the commutativity of multiplicative b-generalized derivations in prime rings. In 2024, Tiwari et al. [11] established results

concerning identities involving $b$- generalized skew derivations in prime rings acting as Jordan derivations. In 2025 Most recently, Fallatah, Shujat, and Ansari [4] proved several results on the commutativity of prime rings admitting $b$-generalized derivations. The present research is motivated by the study of b-generalized derivations satisfying the Strong Commutativity Preserving condition. Furthermore, this work improves upon existing results concerning the commutativity of prime rings that admit b- generalized derivations.

Preliminaries: Throughout the article, the center of an associative ring is denoted by

 

(). If = {0} implies that = 0 or = 0 for each , , then a ring is prime. A derivation is an additive mapping : if, for every, , () = () + ().An additive mapping : is called a generalized derivation, if there exists a derivation : such that () = () + (), for all , . An additive mapping : is called a b-generalized derivation, if there exists a derivation : such that () = () + (), for all

, .

Throughout this paper, we use one of the properties of Martindale right symmetric ring of quotients which states as follows: for any , there exists a dense right ideal such

that and if = 0 or = 0 if and only if = 0. In our case, 0 , we assume that there exists a dense right ideal such that , i.e. or for all and = 0 (or = 0) if and only if = 0. We shall make use of the basic commutator identities: (i) [, ] = [, ] + [, ] ; (ii) [, ] = [, ] + [, ]

(iii) () = () [, ] = () + [, ] ; (iii) () = () [, ] = () + [, ]

Some examples are listed below:

Example 1: Assume is a field of char 2.

11 11

11 0 11

[ 0 0 11 ]. Consequently, is a b-

0 0 0

generalized derivation associated with

1 0 0

derivation and for any fixed = [0 0 0].

0 0 0

Lemma 1[8, Lemma 3]: If a prime ring contains a commutative nonzero right ideal , then is commutative.

Lemma 2[1]: For a prime ring , let be a nonzero left ideal of . If admits a nonzero derivation such that [(), ] () for every , then is commutative.

Lemma 3[2]: For any (), a dense

Suppose that = {[ 0 11 ] \11, 11, 11

right ideal say exists such that = {0}

}. Construct : as () = 11 +

1122 for all . Taking a mapping

: as () = 1122, for all , with derivation , it is evident that is a b- generalized derivation for any fixed . Example 2: Suppose that =

0 11 11

{[0 0 11 ] \11, 11, 11 }, the

0 0 0

mapping , : defined by

( = {0}), then = 0.

Theorem 1: Let be a prime ring, be a nonzero the dense ideal of and : be a b-generalized derivation associated with derivation : , if [, ] = 0, for all

, , then is commutative.

Proof: According to the hypothesis, [, ] = 0, for all , (1)

We replacing by in equation (1), we get

Using equation (1) in the above equation,

is a b-generalized derivation with derivation

and for any fixed .

11				0
Example	3:	Let	= {[ 0	0
			0	0

We replacing by in the above equation, we get

Using equation (2) in the above equation, we get

11

defined by ([

We replacing by in the equation (3),

The equation (3) from right multiply by , we get

Subtracting equation (4) from equation (5), we get

We replacing by in the above equation, we get

Since is prime, we get [, ] = 0 or

[(), ] = 0, for all , , , . Using set theoretic notation, we can construct two proper subsets of namel

Applying, Brauers theorem clearly each of

and is additive subgroup of such that =

. But, a group cannot be the set-theoretic union of its two proper subgroups. Hence = or = .

In the first case = , that is [, ] = 0,

for all , . (7)

We replace by in the equation (7), we get

[, ] = 0, for all , , .

[, ] + [, ] = 0, for all , ,

Using equation (7), we get

[, ] = 0, for all , , .

We replace by in the above equation, we get

[, ] = 0, for all , , , .

[, ] = 0, for all , , .

The prime condition we get, = 0 or

[, ] = 0, for all , and .

Using Lemma 3, we met a contradiction in the case = 0, for all .

Therefore, [, ] = 0, for all and .

We replace by in the above equation, we get

[, ] = 0, for all and , . [, ] = 0, for all and , .

Discard the case = 0 and apply the primeness condition, we find [, ] = 0 for all

and . Using Lemma1, we get is commutative.

In the second case = , that is [(), ] =

0, for all , , , .

We replace by in the above equation, we get

[(), ] = 0, for all , , , . Using primeness condition, we get either = 0 or [(), ] = 0, for all , .

But 0, we get [(), ] = 0, for all , .

This indicates that is commuting derivation on that is not zero. Accordingly, is commutative employing Lemma 2. Thus, the proof is completed.

Theorem 2: Let be a prime ring, be a nonzero the dense ideal of and : be a b-generalized derivation associated with derivation : , if [, ] () = 0, for all , , then is commutative.

Proof: we consider the case

[, ] () = 0, for all , . (8) We replacing by in equation (8), we get [ , ] ( ) = 0

Using equation (8) in the above equation, we get

[, ]() = 0, for all , (9)

Using the same arguments after (2) in the proof of theorem 1, we get the required result. By the similar approach, we can prove the same conclusion holds for [, ] + () = 0, for all , .

Theorem 3: Let be a prime ring, be a nonzero the dense ideal of and : be a b-generalized derivation associated with derivation : , if [, ] [, ] = 0, for all , , then is commutative.

Proof: we consider the case

We replacing by in equation (10), we get

Using equation (10) in the above equation, we get

Using the same arguments after (2) in the proof of theorem 1, we get the required result. By the similar approach, we can prove the same conclusion holds for [, ] + [, ] = 0, for all , .

References

1. Bell, H. E. and Martindale III, W. S., Centralizing mappings on semiprime rings, Canad. Math. Bull., Vol. 30, No. 1, pp. 92-101, 1987
2. Beidar, K. I., Martindale III, W. S. and Mikhalev, A. V., Rings with generalized identities, Monographs and Textbooks in Pure and Applied Mathematics, 196, Marcel Dekker Inc., New York, 1996.
3. De Filippis, V. and Wei, F., b-generalized skew derivation on Lie ideals, Mediterr. J. Math. 15 (2), pp.1-24, 2018
4. Fallatah, A., Shujat,F. and Ansari, A.Z., Commutativity of Prime Rings with b-generalized Derivations Mathematics and Statistics 13(4): 228-231, 2025, DOI: 10.13189/ms.2025.130406
5. Kosan, M. T. and Lee, T. K., b-Generalized derivations of semiprime rings having nilpotent values, J. Austrl. Math. Soc., Vol. 96, No. 3, pp. 326-337, 2014.DOI: 10.1017/S1446788713000670
6. Liu, C.K., An Engle condition with b-generalized derivations, Linear Multilinear Algebra 65 (2), 300-312,

   2017.

7. Liau, P. K. and Liu, C. K., An Engle condition with b- generalized derivations for Lie ideals, J.AlgebraAPP.17(3),1850046(17Pages) 2018.

   DOI:10.1142/S0219498818500469

8. Mayne, J. H., Centralizing automorphisms of prime rings,

   Canad. Math. Bull., Vol. 19, pp. 113-115, 1976.

   DOI:https://doi.org/10.4153/CMB-1976-017-1

9. Mozumder, M., Abbasi, A., Dar. N.A., Fosner, A. and Khan,

   M. S., A note on b-generalized derivations in rings with involution, Italian Journal of Pure and Applied Mathematics, Vol. 49, pp. 482-489, 2023.

10. Mozumder, M., Ahmed, W., Raza, M. A. and Abbasi, A., Commutativity of multiplicative b-generalized derivations of prime rings, Korean J. Math., Vol. 31, No. 1, pp. 95-107, 2023. DOI: 10.11568/kjm.2023.31.1.95
11. Tiwari, S.K. and Gupta, P.,Identities involving b- generalized skew derivations in prime rings acting as Jordan derivation, Communications in Algebra, Vol. 52, No. 11, pp. 4612-4630, 2024. DOI:

10.1080/00927872.2024.2354425