The k-Derivation of a Gamma-Ring

Authors : Hatice KANDAMAR

Pages : 221-231

View : 9 | Download : 6

Publication Date : 0000-00-00

Article Type : Research Paper

Abstract :In this paper, the k-derivation is defined on a G-ring M insert ignore into journalissuearticles values(that is, if M is a G-ring, d:M\to M and k:G\to G are to additive maps such that dinsert ignore into journalissuearticles values(ab b );= dinsert ignore into journalissuearticles values(a);b b + akinsert ignore into journalissuearticles values(b);b + ab dinsert ignore into journalissuearticles values(b); for all a,b\in M, \quad b \in G, then d is called a k-derivation of M); and the following results are proved. insert ignore into journalissuearticles values(1); Let R be a ring of characteristic not equal to 2 such that if xry=0 for all x, y\in R then r=0. If d is a k-derivation of the insert ignore into journalissuearticles values(R=);G-ring R with k=d, then d is the ordinary derivation of R. insert ignore into journalissuearticles values(2); Let M be a nonzero prime G-ring of characteristic not equal to 2, g be an element of G and a is an element in M such that [ [x, a]g , a]g =0 for all x\in M. Then ag a = 0 or a\in Cg. insert ignore into journalissuearticles values(3); Let M be a prime G-ring with CharM \ne 2, d be a nonzero k-derivation of M, g be a nonzero element of G and kinsert ignore into journalissuearticles values(g); \ne 0. If dinsert ignore into journalissuearticles values(M); \subseteq Cg, then M is a commutative G-ring.
Keywords : k derivation, derivation, commutativity, gamma ring