[PDF] Top 20 Identities with derivations and automorphisms on semiprime rings

... 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 1.3]. ... See full document

... of automorphisms to von Neumann algebras, not necessarily abelian, by using implicitly the dependent elements of an ...of automorphisms were later studied by Choda et ...of automorphisms in the ... See full document

... A well-know result of Herstein [11] states that every Jordan isomorphism on a prime ring of characteristic di ff erent from 2 is either an isomorphism or an anti-isomorphism. We remark that the situation where the ... See full document

... and semiprime rings admitting derivations and generalized derivations which satisfy appropriate algebraic conditions on suitable subset of the ...a semiprime ring with a nonzero ideal I ... See full document

... P ROOF : Since any derivation d can be uniquely extended to a derivation in U , and U and R satisfy the same differential identities ( see [20]), then ad(x) ∈ Z(U ), for all x ∈ U . Let P ∈ M(C) such that U/P U is ... See full document

... for all x, y ∈ U . Let M (C) be the set of all maximal ideals of C and P ∈ M (C). Now by the standard theory of orthogonal completions for semiprime rings (see [24, p.31-32]), we have P U is a prime ideal ... See full document

... Let R be an associative ring. The center of R is denoted by Z. 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. We will make extensive use of ... See full document

... Abstract Let R be an associative ring. A mapping f : R −→ R is said to be additive if f (x + y) = f (x) + f (y) holds for all x, y ∈ R. An additive mapping d : R −→ R is called a derivation if d(xy) = d(x)y + xd(y) holds ... See full document

... Z (R). Let n be any integer. A ring R is said to be n- torsion free, if for all x, y  R, n x = 0  x = 0. Recall that a ring R is said to be prime, if a R b = 0,  a, b  R  either a=0 (or) b=0 and a ring R is said to ... See full document

... in semiprime rings there are no nonzero nilpotent dependent elements (see ...of automorphisms of abelian von Neumann algebras of Murray and von Neumann [14, ...to derivations and ... See full document

... the identities [xy, z] = x[y, z] + [x, z]y and [x, yz] = y[x, z] + [x, y]z for all x, y, z ∈ ...is semiprime if aRa = (0) implies a = 0 and it is prime if aRb = (0) implies a = 0 or b = ...is ... See full document

... a semiprime -ring, if ΓΓ = 0 implies = 0, for all ∈ ...commutator identities: [, %] & = [, %] & + [, %] & and [, %] & = [, ] & % + [, %] & , for all ... See full document

... commentator identities given by [xβy, z]α = xβ[y, z]α + [x, z]αβy + x[β, α]zy & [x, yβz]α = yβ[x, z]α + [x, y]αβz+ y[β, α]xz, for all x, y, z ∈ M and for all α, β ∈ ...commentator identities reduce to ... See full document

... a semiprime ring R. They proved that if R is a semiprime ring and dx, x, y 0 for all x in a non-zero left ideal of R and y ∈ R, then R contains a non-zero central ... See full document

... The derivations in Γ-near-rings have been introduced by Bell and Mason ...of derivations in ...with derivations. Some characterizations of Γ-near-rings and regularity conditions were ... See full document

... Throughout the present paper, unless otherwise mentioned, N will denote a left near- ring. N is called a prime near-ring if xN y = {0} implies x = 0 or y = 0. It is called semiprime if xN x = {0} implies x = 0. ... See full document

... The notion of Γ-ring was introduced as a generalized extension of the concept on classical ring. From its first appearance, the extensions and the generalizations of various important results in the theory of classical ... See full document

... free semiprime Г- ring such that 𝑥𝛼𝑦𝛽𝑧 = 𝑥𝛽𝑦𝛼𝑧 for all α, βϵГ , x, y, zϵM and two symmetric higher bi-derivations 𝑑 𝑛 and 𝑔 𝑛 for all n𝜖N are orthogonal if and only if 𝑑 𝑛 (𝑥, 𝑦)𝛼𝑔 𝑛 (𝑦, 𝑧) + 𝑔 𝑛 (𝑥, 𝑦)𝛼𝑑 𝑛 ... See full document

... antiautomorphism of ℛ. A ring ℛ is said to be 𝓃- torsion free if 𝓃𝒶=0 with 𝒶∈ℛ then 𝒶=0, where 𝓃 is nonzero integer (1). For any 𝜐,𝛾∈ℛ, the commutator 𝜐𝛾-𝛾𝜐 is denoted by [𝜐, 𝛾] (2). Recall that a ring ℛ is said to be ... See full document

... free semiprime Γ -ring satisfying the assumption (A) and if T : M → M is an additive mapping such that T ( x α y β x ) = x α T ( y ) β x for all x , y ∈ M and α , β ∈ Γ , then T is a ... See full document