Generalized Derivations of Prime Rings
Let R be an associative prime ring, U a Lie ideal such that u2∈U for all u∈U. An additive function F:R→R is called a generalized derivation if there exists a derivation d:R→R such that F(xy)=F(x)y+xd(y) holds for all x,y∈R. In this paper, we prove that d=0 or U⊆Z(R) if any one of the following con...
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2007-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2007/85612 |
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| author | Huang Shuliang |
| author_facet | Huang Shuliang |
| author_sort | Huang Shuliang |
| collection | DOAJ |
| description | Let R be an associative prime ring, U a Lie ideal such
that u2∈U for all u∈U. An additive function F:R→R is
called a generalized derivation if there exists a derivation d:R→R such that F(xy)=F(x)y+xd(y) holds
for all x,y∈R. In this paper, we prove that d=0 or U⊆Z(R) if any one of the following conditions holds: (1) d(x)∘F(y)=0, (2) [d(x),F(y)=0], (3) either d(x)∘F(y)=x∘y or d(x)∘F(y)+x∘y=0, (4) either d(x)∘F(y)=[x,y] or d(x)∘F(y)+[x,y]=0,
(5) either d(x)∘F(y)−xy∈Z(R) or d(x)∘F(y)+xy∈Z(R), (6) either [d(x),F(y)]=[x,y] or [d(x),F(y)]+[x,y]=0,
(7) either [d(x),F(y)]=x∘y or [d(x),F(y)]+x∘y=0 for all x,y∈U. |
| format | Article |
| id | doaj-art-77f56175dc284b79b3aaf9c975069e7a |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2007-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-77f56175dc284b79b3aaf9c975069e7a2025-08-20T03:35:36ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252007-01-01200710.1155/2007/8561285612Generalized Derivations of Prime RingsHuang Shuliang0Department of Mathematics, Chuzhou University, Chuzhou 239012, ChinaLet R be an associative prime ring, U a Lie ideal such that u2∈U for all u∈U. An additive function F:R→R is called a generalized derivation if there exists a derivation d:R→R such that F(xy)=F(x)y+xd(y) holds for all x,y∈R. In this paper, we prove that d=0 or U⊆Z(R) if any one of the following conditions holds: (1) d(x)∘F(y)=0, (2) [d(x),F(y)=0], (3) either d(x)∘F(y)=x∘y or d(x)∘F(y)+x∘y=0, (4) either d(x)∘F(y)=[x,y] or d(x)∘F(y)+[x,y]=0, (5) either d(x)∘F(y)−xy∈Z(R) or d(x)∘F(y)+xy∈Z(R), (6) either [d(x),F(y)]=[x,y] or [d(x),F(y)]+[x,y]=0, (7) either [d(x),F(y)]=x∘y or [d(x),F(y)]+x∘y=0 for all x,y∈U.http://dx.doi.org/10.1155/2007/85612 |
| spellingShingle | Huang Shuliang Generalized Derivations of Prime Rings International Journal of Mathematics and Mathematical Sciences |
| title | Generalized Derivations of Prime Rings |
| title_full | Generalized Derivations of Prime Rings |
| title_fullStr | Generalized Derivations of Prime Rings |
| title_full_unstemmed | Generalized Derivations of Prime Rings |
| title_short | Generalized Derivations of Prime Rings |
| title_sort | generalized derivations of prime rings |
| url | http://dx.doi.org/10.1155/2007/85612 |
| work_keys_str_mv | AT huangshuliang generalizedderivationsofprimerings |