# On Semigroup Ideals and Right n-Derivation in 3-Prime Near-Rings

## DOI:

https://doi.org/10.21123/bsj.2023.8086## Keywords:

Generalized right derivations, Prime near-ring, Right derivations, Right n –derivations, Semigroup ideals## Abstract

The current paper studied the concept of right n-derivation satisfying certified conditions on semigroup ideals of near-rings and some related properties. Interesting results have been reached, the most prominent of which are the following: Let M be a 3-prime left near-ring and A_1,A_2,…,A_n are nonzero semigroup ideals of M, if d is a right n-derivation of M satisfies on of the following conditions,

d(u_1,u_2,…,(u_j,v_j ),…,u_n )=0 ∀ 〖 u〗_1 〖ϵA〗_1 ,u_2 〖ϵA〗_2,…,u_j,v_j ϵ A_j,…,〖u_n ϵA〗_u;

d((u_1,v_1 ),(u_2,v_2 ),…,(u_j,v_j ),…,(u_n,v_n ))=0 ∀u_1,v_1 〖ϵA〗_1,u_2,v_2 〖ϵA〗_2,…,u_j,v_j ϵ A_j,…,〖u_n,v_n ϵA〗_u ;

d((u_1,v_1 ),(u_2,v_2 ),…,(u_j,v_j ),…,(u_n,v_n ))=(u_j,v_j ) ∀ u_1,v_1 〖ϵA〗_1 ,u_2,v_2 〖ϵA〗_2,…,u_j,v_j ϵ A_j,…,〖u_n,v_n ϵA〗_u;

If d+d is an n -additive mapping from A_1×A_2×…×A_n to M;

d(u_1,u_2,…,(u_j,v_j ),…,u_n)∈ Z(M) ∀〖 u〗_1 〖ϵA〗_1 ,u_2 〖ϵA〗_2,…,u_j,v_j ϵ A_j,…,〖u_n ϵA〗_u,;

d((u_1,v_1 ),(u_2,v_2 ),…,(u_j,v_j ),…,(u_n,v_n ))∈ Z(M) ∀ u_1,v_1 〖ϵA〗_1 ,u_2,v_2 〖ϵA〗_2,…,u_j,v_j ϵ A_j,…,〖u_n,v_n ϵA〗_u;

Then M is a commutative ring.

Received 10/11/2022

Revised 24/12/2022

Accepted 26/12/2022

Published Online First 20/5/2023

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References

Pilz G. Near-Rings. 2nd ed. North Holland: American Elsevier, Amsterdam; 1983. 541 p.

Bell H. On Derivations in Near-Rings II. Near-rings, Near-ﬁelds and k-loops. Kluwer Academic Publishers. Dordrecht. 1997; 426: 191–197

Boua A, Farhan E. Generalized derivations on near-rings with identities. J. Discret. Math. Sci. Cryptogr. 2021; 26(4):1-14.

Enguady A, Boua A, Farhan E. Some Identities of 3-Prime Near-Rings Involving Jordan Ideals and Left Generalized Derivations. Iraqi J. Sci. 2021; 62(6): 1961-1967

Boua A and Farhan E. Generalized homoderivations in near-rings. Indian J. Math. 2021; 63(2): 229-242.

Ashraf A, Siddeeque M. On permuting n-derivations in near-rings. Commun. Kor. Math. Soc. 2013; 28(4): 697–707.

Farhan E. A Study on n-Derivation in Prime Near – Rings. Iraqi J. Sci. 2020; 61(3): 620-624.

Farhan E. Generalized n-Derivation in Prime Near – Rings. Sci. Int. (Lahore). 2018; 30(6): 865-868

Farhan E. α-N-derivations in prime near – rings. AIP Conf Proc. 2019; 2201(1): 020011.

. Majeed A, Farhan E. Right n-derivations in prime Near–Rings. Journal Al-Qadisiyah / Pure Sciences. 2018; 2(3): 31-41.

Mouhssine S, Boua A, Farhan E. On Some Identities For Right n-Derivatons In 3-Prime Near-Rings. Indian Journal. 2022; 64(2): 161-194..

Farhan. E. Near – Rings with Generalized Right –Derivations. ijs. 2021; 62(7): 2334-2342.

Mahmood AH. Notes on Traces of a Symmetric Generalized (σ, τ)-Biderivations and Commutativity in Prime Rings, Baghdad Sci. J. 2018; 14(1): 213-218

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