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We characterize bilinear functionals ϕ on a symmetric algebra A satisfying the two-sided zero product property (the 2-zpp, i.e., ϕ (x , y) = 0 whenever x y = y x = 0). If A is also a zero product determined algebra and if every derivation of the algebra A is inner, then A is a 2-zpd algebra (i.e., every bilinear functional on A satisfying the 2-zpp is of the form (x , y) ↦ τ 1 (x y) + τ 2 (y x) for x , y ∈ A , where τ 1 , τ 2 are linear functionals on A). Conversely, if A is a finite-dimensional 2-zpd algebra, then the derivations of A are characterized, that is, given any derivation d of the algebra A , there exists a ∈ A such that, for all x ∈ A , d (x) − [ a , x ] lies in the Jacobson radical of A. Finally, we determine all bilinear functionals satisfying the 2-zpp on a specific zpd symmetric algebra and hence decide whether such an algebra is 2-zpd. [ABSTRACT FROM AUTHOR]

Let R be a prime ring with center Z ( R ) and with extended centroid C . We give a complete characterization of Jordan derivations of R when char R = 2 and dim C ⁡ R C = 4 : An additive map δ : R → R C is a Jordan derivation if and only if there exist a derivation d : R → R C and an additive map μ : R → C such that δ = d + μ and μ ( x 2 ) = 0 for all x ∈ R . As consequences, it is proved among other things: Any Z ( R ) -linear Jordan derivation of R is a derivation if dim C ⁡ R C < ∞ . Moreover, if C is either a finite field or an algebraically closed field, where char C = 2 and n ≥ 2 , then every Jordan derivation of M n ( C ) is a derivation. [ABSTRACT FROM AUTHOR]

Abstract: Let D be a division algebra finite-dimensional over its center C, , the matrix algebra over D, and V be a vector space over C. We characterize all n-linear forms on Ω in terms of reduced traces and elementary operators. For , it is proved that a bilinear form vanishes on zero products of xy and yx if and only if there exist linear maps such that for all . As an application, a bilinear form B is completely characterized if whenever satisfy , where ξ is a fixed nonzero element in C. [Copyright &y& Elsevier]

Abstract: A ring is called nil-clean if each of its elements is a sum of an idempotent and a nilpotent. In response to a question of S. Breaz et al. in [1], we prove that the matrix ring over a division ring D is a nil-clean ring if and only if . As consequences, it is shown that the matrix ring over a strongly regular ring R is a nil-clean ring if and only if R is a Boolean ring, and that a semilocal ring R is nil-clean if and only if its Jacobson radical is nil and is a direct product of matrix rings over . [Copyright &y& Elsevier]

Abstract: Let D be a division algebra finite-dimensional over its center C and let , the matrix ring over D. By the length of a linear generalized polynomial (GP) , we mean the least positive integer n such that can be represented in the form for some . We denote by the length of . By a central linear GP for A we mean a nonzero linear GP with central values on A. In this paper we characterize all central linear GPs for A and determine the lengths of all central linear GPs for A. [Copyright &y& Elsevier]

Abstract: A new characterization of von Neumann regular rings is obtained, in terms of simple 0-multiplication of matrices, and is used to establish the natural connections between von Neumann regular rings and feebly Baer modules and rings. [Copyright &y& Elsevier]

Abstract: Let be a -algebra acting densely on , where is a commutative ring with unity and is a right vector space over a division -algebra . Let be a nonzero right ideal of and let be a nonzero polynomial over with constant term 0 such that for some coefficient of . Suppose that is a nonzero derivation. It is proved that if for all and for some positive integer , then either is generated by an idempotent of finite rank or for some of finite rank. In addition, if is multilinear, then can be chosen such that . [Copyright &y& Elsevier]

Abstract: Let be a -algebra acting densely on , where is a commutative ring with unity and is a right vector space over a division -algebra . Let be an arbitrary and fixed polynomial over in noncommuting indeterminates with constant term such that for some occurring in the coefficients of . It is proved that a right ideal of is generated by an idempotent of finite rank if and only if the rank of is bounded above by a same natural number for all . In this case, the rank of the idempotent that generates is also explicitly given. The results are then applied to considering the triangularization of and the irreducibility of , where denotes the additive subgroup of generated by the elements for . [Copyright &y& Elsevier]

Abstract: Let be a prime ring with extended centroid and let be a reduced differential polynomial with coefficients in , the symmetric Martindale quotient ring of , and with zero constant term. Let and . We prove that the finiteness of and the finite-dimensionality of the -span of are equivalent to that of and that of the -span of , respectively. Hence some questions on differential polynomials are reduced to those on ordinary generalized polynomials. Let and be two derivations of a Lie ideal of and a right ideal of . As applications of our theorems, we obtain the necessary and sufficiency conditions for the finiteness of and and for the finite-dimensionality of the -spans of and . [Copyright &y& Elsevier]

Abstract: We prove a Wedderburn–Artin type theorem for algebraic prime subalgebras in simple Artinian algebras, giving a generalized version of Yahaghi’s theorem [B.R. Yahaghi, On F-algebras of algebraic matrices over a subfield F of the center of a division ring, Linear Algebra Appl. 418 (2006) 599–613]. We also show that every semiprime left algebraic subring in a semiprime right Goldie ring must be a semiprime Artinian ring. [Copyright &y& Elsevier]

Abstract: Let R be a prime ring with extended centroid C. Let a ∈ R be C-algebraic. We compute the minimal polynomial of the inner derivation ad(a) (resp. the inner automorphism I a for unit a) in terms of the minimal polynomial of a. Chung and Luh proved that the index of nilpotency of a nilpotent derivation is an odd number if char R ≠2 and is a 2-power if char R =2. As an application, we generalize this to arbitrary algebraic derivations and automorphisms. [Copyright &y& Elsevier]