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AI InsightIf d(M) \subseteq Cg, then M is a commutative G-ring.
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Özet
In this paper, the k-derivation is defined on a G-ring M (that is, if M is a G-ring, d:M\to M and k:G\to G are to additive maps such that d(ab b )= d(a)b b + ak(b)b + ab d(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. (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 (R=)G-ring R with k=d, then d is the ordinary derivation of R. (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. (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 k(g) \ne 0. If d(M) \subseteq Cg, then M is a commutative G-ring.
Abstract
In this paper, the k-derivation is defined on a G-ring M (that is, if M is a G-ring, d:M\to M and k:G\to G are to additive maps such that d(ab b )= d(a)b b + ak(b)b + ab d(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. (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 (R=)G-ring R with k=d, then d is the ordinary derivation of R. (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. (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 k(g) \ne 0. If d(M) \subseteq Cg, then M is a commutative G-ring.