A note on derivations in semiprime rings
We prove in this note the following result. Let n>1 be an integer and let R be an n!-torsion-free semiprime ring with identity element. Suppose that there exists an additive mapping D:R→R such that D(xn)=∑j=1nxn−jD(x)xj−1 is fulfilled for all x∈R. In this case, D is a derivation. This research is...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
2005-01-01
|
| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/IJMMS.2005.3347 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1832556407925243904 |
|---|---|
| author | Joso Vukman Irena Kosi-Ulbl |
| author_facet | Joso Vukman Irena Kosi-Ulbl |
| author_sort | Joso Vukman |
| collection | DOAJ |
| description | We prove in this note the following result. Let n>1 be an integer and let R be an n!-torsion-free semiprime ring with identity element. Suppose that there exists an additive mapping D:R→R such that D(xn)=∑j=1nxn−jD(x)xj−1 is fulfilled for all x∈R. In this case, D is a derivation. This research is motivated by the work of Bridges and Bergen (1984). Throughout, R will represent an associative ring with center Z(R). Given an integer n>1, a ring R is said to be n-torsion-free if for x∈R, nx=0 implies that x=0. Recall that a ring R is prime if for a,b∈R, aRb=(0) implies that either a=0 or b=0, and is semiprime in case aRa=(0) implies that a=0. An additive mapping D:R→R is called a derivation if D(xy)=D(x)y+xD(y) holds for all pairs x,y∈R and is called a Jordan derivation in case D(x2)=D(x)x+xD(x) is fulfilled for all x∈R. Every derivation is a Jordan derivation. The converse is in general not true. A classical result of Herstein (1957) asserts that any Jordan derivation on a prime ring with characteristic different from two is a derivation. A brief proof of Herstein's result can be found in 1988 by Brešar and Vukman. Cusack (1975) generalized Herstein's result to 2-torsion-free semiprime rings (see also Brešar (1988) for an alternative proof). For some other results concerning derivations on prime and semiprime rings, we refer to Brešar (1989), Vukman (2005), Vukman and Kosi-Ulbl (2005). |
| format | Article |
| id | doaj-art-bb06b71826694554bb039e9a1dcb2c73 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2005-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-bb06b71826694554bb039e9a1dcb2c732025-02-03T05:45:33ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252005-01-012005203347335010.1155/IJMMS.2005.3347A note on derivations in semiprime ringsJoso Vukman0Irena Kosi-Ulbl1Department of Mathematics, Faculty of Education, University of Maribor, Koroška Cesta 160, Maribor 2000, SloveniaDepartment of Mathematics, Faculty of Education, University of Maribor, Koroška Cesta 160, Maribor 2000, SloveniaWe prove in this note the following result. Let n>1 be an integer and let R be an n!-torsion-free semiprime ring with identity element. Suppose that there exists an additive mapping D:R→R such that D(xn)=∑j=1nxn−jD(x)xj−1 is fulfilled for all x∈R. In this case, D is a derivation. This research is motivated by the work of Bridges and Bergen (1984). Throughout, R will represent an associative ring with center Z(R). Given an integer n>1, a ring R is said to be n-torsion-free if for x∈R, nx=0 implies that x=0. Recall that a ring R is prime if for a,b∈R, aRb=(0) implies that either a=0 or b=0, and is semiprime in case aRa=(0) implies that a=0. An additive mapping D:R→R is called a derivation if D(xy)=D(x)y+xD(y) holds for all pairs x,y∈R and is called a Jordan derivation in case D(x2)=D(x)x+xD(x) is fulfilled for all x∈R. Every derivation is a Jordan derivation. The converse is in general not true. A classical result of Herstein (1957) asserts that any Jordan derivation on a prime ring with characteristic different from two is a derivation. A brief proof of Herstein's result can be found in 1988 by Brešar and Vukman. Cusack (1975) generalized Herstein's result to 2-torsion-free semiprime rings (see also Brešar (1988) for an alternative proof). For some other results concerning derivations on prime and semiprime rings, we refer to Brešar (1989), Vukman (2005), Vukman and Kosi-Ulbl (2005).http://dx.doi.org/10.1155/IJMMS.2005.3347 |
| spellingShingle | Joso Vukman Irena Kosi-Ulbl A note on derivations in semiprime rings International Journal of Mathematics and Mathematical Sciences |
| title | A note on derivations in semiprime rings |
| title_full | A note on derivations in semiprime rings |
| title_fullStr | A note on derivations in semiprime rings |
| title_full_unstemmed | A note on derivations in semiprime rings |
| title_short | A note on derivations in semiprime rings |
| title_sort | note on derivations in semiprime rings |
| url | http://dx.doi.org/10.1155/IJMMS.2005.3347 |
| work_keys_str_mv | AT josovukman anoteonderivationsinsemiprimerings AT irenakosiulbl anoteonderivationsinsemiprimerings AT josovukman noteonderivationsinsemiprimerings AT irenakosiulbl noteonderivationsinsemiprimerings |