Daniel Eremita

On certain equations in rings

In this paper we prove the following result: Let R be a 2-torsion free semiprime ring. Suppose there exists an additive mapping T : R → R such that T(xyx) = T(x)yx − xT(y)x + xyT(x) holds for all pairs x, y ∈ R . Then T is of the form 2T(x) = qx + xq , where q is a fixed element in the symmetric Martindale ring of quotients of R .

Functional identities of degree 2 in triangular rings revisited

Using the notion of the maximal left ring of quotients, our recent result on the solutions of functional identity in triangular rings is generalized. Consequently, generalizations of known results on commuting additive maps and generalized inner biderivations of triangular rings are obtained.

Functional Identities in Jordan Algebras: Associating Traces and Lie Triple Isomorphisms

Abstract Let J be a prime nondegenerate Jordan algebra. The problem of describing the form of a symmetric biadditive map B : J × J → J satisfying (B(a, a) · b) · a = B(a, a) · (b · a) for all a, b ∈ J is discussed. As an application, Lie triple isomorphisms of Jordan algebras are considered.

The Lower Socle and Finite Rank Elementary Operators

Abstract We define the lower socle of a semiprime algebra 𝒜 as the sum of all minimal left ideals 𝒜e where e is a minimal idempotent such that the division algebra e𝒜e is finite dimensional. We study the connection between the condition that the elements a k , b k , 1 ≤ k ≤ n, lie in the lower socle of 𝒜 and the condition that the elementary operator x ↦ a 1 xb 1 + ċ + a n xb n has finite rank. As an application we obtain some results on derivations certain of whose powers have finite rank.

On additivity of centralisers

Let R be a ring and let M be a bimodule over R . We consider the question of when a map φ: R → M such that φ( ab ) = φ (a)b for all a, b ∈ R is additive.

Biderivations on tensor products of algebras

ABSTRACT Let A be a noncommutative unital prime algebra and let S be a commutative unital algebra over a field 𝔽. We describe the form of biderivations of the algebra A⊗S. As an application, we determine the form of commuting linear maps of A⊗S.