[PDF] Top 20 On the Lie ring of derivations of a semiprime ring

... Throughout the text R stands for an associative ring (possibly without identity) and n for a positive integer. By Z(R) we denote the center of R. The ring R is called semiprime, if it has no nonzero ... See full document

... Let R be an associative ring. The centre of R is denoted by Z (R). For x, y ∈ R, the symbol [x, y] will denote the commutator xy − yx and the symbol x ◦ y will denote the anticommutator xy + yx. Recall that a ... See full document

... Lemma 1 [11, Lemma 1]. Let R be a semiprime ring. Suppose that the relation axb + bxc = 0 holds for all x ∈ R and some a,b,c ∈ R. In this case, (a + c)xb = 0 is satisfied for all x ∈ R. Lemma 2 [12, Lemma ... See full document

... of derivations on semiprime ...a semiprime ring with center Z(R), and let f , g be derivations of R such that f (x)x + xg(x) ∈ Z(R) for all x ∈ R, then f and g are ...linear ... See full document

... We call an element element a ∈ R a dependent element of a derivation d : R → R if dxa a, xa holds for all x ∈ R. Following 3, a derivation d : R → R is said to act freely on R or a free action in case zero is the only ... See full document

... Every ring left derivation resp., ring derivation on ring is a Jordan left derivation ...resp., ring Jordan ...a ring Jordan left derivation on a 2-torsion free prime ring is a ... See full document

... for all x, y ∈ I. Right multiplying (11) by y and then subtracting from (12), we have F(x)[F(y), y] = 0 for all x, y ∈ I. Putting xz for x in the last relation, we obtain F (x)z[F (y), y] = 0 for all x, y, z ∈ I. This ... See full document

... a ring with center Z and I a nonzero ideal of ...the semiprime ring R must contains a nonzero central ideal, provided d I / ...prime ring, R must be commutative, provided d / ... See full document

... Fact 2.2. [Lee, 1999, Theorem 4] Let R be a semiprime ring. Then every gen- eralized derivation F on a dense right ideal of R is uniquely extended to U and assumes the form F (x) = ax + d(x) for some a ∈ U ... See full document

... associative ring with center Z (R). We de- note by Q and C the maximal ring of quotient and the extended centroid of a prime ring, ...A ring is said to 2-torsion free if 2x = 0 (where x ∈ R) ... See full document

... also derivations on rings can tell us about the structure of the ...a ring is commutative if and only if the only inner derivation on the ring is ...Also derivations can be helpful for ... See full document

... Proof. Assume either d(I) 6= 0 or g(I) 6= 0. Thus, by Theorem 1.9 respectively we have that either [d(R), I] = (0) or [g(R), I] = (0). In any case, again by [13], R must contain some non-zero central ideals. Corollary ... See full document

... λ ∈ K . Ten years ago, Y. Stein posed a problem of describing all weakly semisimple polynomials (such a description would character- ize all two dimensional nonabelian subalgebras of the Lie algebra of all ... See full document

... Now, the following lemmas of orthogonal symmetric higher bi-derivations on Г- ring M. Lemma (2-3):(2) Let M be a 2-torsion free semiprime Г-ring an x,y be elements of M. If for all 𝛼, 𝛽𝜖Г, ... See full document

... a ring a concept more general than a ring was definedby Nobusawa ...near ring has been introduced by Bell and ...of derivations in near ...and semiprime rings admitting suitably ... See full document

... prime ring of characteristic di ff erent from 2 is either an isomorphism or an ...are semiprime rings does not ...prime ring of characteristic dif- ferent from 2 is a derivation ... See full document

... centralizing derivations like J. Vukman who investigated symmetric bi-derivations on prime and semiprime rings ...and semiprime rings (10) and more results in (11, 12, 13, 14, ...of Lie ... See full document

... The ring M is n-torsion-free if nx 0, x ∈ M implies x 0, where n is a positive integer, M is prime if aΓMΓb 0 implies a 0 or b 0, and it is semiprime if aΓMΓa 0 implies a ... See full document

... The concepts of sub ring and ideal are imitated from the classical case. Throughout the paper, M denotes a Γ-ring with center Z(M). If 2x = 0 for x ∈ M implies x = 0A ring, then M is said to be ... See full document

... Recently in [26], Rehman and Raza proved the following: Let R be a prime ring of char (R) 6= 2, Z the center of R, and L a nonzero Lie ideal of R. If R admits a generalized derivation F associated with a ... See full document