On Semigroup Ideals and Right n-Derivation in 3-Prime Near-Rings
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 n...
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University of Baghdad, College of Science for Women
2024-01-01
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| Series: | مجلة بغداد للعلوم |
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| Online Access: | https://bsj.uobaghdad.edu.iq/index.php/BSJ/article/view/8086 |
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| author | Enaam Farhan |
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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.
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| format | Article |
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| issn | 2078-8665 2411-7986 |
| language | English |
| publishDate | 2024-01-01 |
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| series | مجلة بغداد للعلوم |
| spelling | doaj-art-2addeef2aa4d4c99978a33d21a6f72a92025-08-20T03:14:35ZengUniversity of Baghdad, College of Science for Womenمجلة بغداد للعلوم2078-86652411-79862024-01-0121110.21123/bsj.2023.8086On Semigroup Ideals and Right n-Derivation in 3-Prime Near-RingsEnaam Farhan0Department of Supervisory Specialization, Directorate General of Education in AL-Qadisiyah, AL-Qadisiyah, Iraq. 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. https://bsj.uobaghdad.edu.iq/index.php/BSJ/article/view/8086Generalized right derivations, Prime near-ring, Right derivations, Right n –derivations, Semigroup ideals |
| spellingShingle | Enaam Farhan On Semigroup Ideals and Right n-Derivation in 3-Prime Near-Rings مجلة بغداد للعلوم Generalized right derivations, Prime near-ring, Right derivations, Right n –derivations, Semigroup ideals |
| title | On Semigroup Ideals and Right n-Derivation in 3-Prime Near-Rings |
| title_full | On Semigroup Ideals and Right n-Derivation in 3-Prime Near-Rings |
| title_fullStr | On Semigroup Ideals and Right n-Derivation in 3-Prime Near-Rings |
| title_full_unstemmed | On Semigroup Ideals and Right n-Derivation in 3-Prime Near-Rings |
| title_short | On Semigroup Ideals and Right n-Derivation in 3-Prime Near-Rings |
| title_sort | on semigroup ideals and right n derivation in 3 prime near rings |
| topic | Generalized right derivations, Prime near-ring, Right derivations, Right n –derivations, Semigroup ideals |
| url | https://bsj.uobaghdad.edu.iq/index.php/BSJ/article/view/8086 |
| work_keys_str_mv | AT enaamfarhan onsemigroupidealsandrightnderivationin3primenearrings |