Algebra ICM 2006 – Posters. Abstracts. Section 02 On generalized left derivations in rings.pdf

ICM 2006 Posters Abstracts Section 02 Algebra ICM 2006 – Posters. Abstracts. Section 02 On generalized left derivations in rings Shakir Ali Department of Mathematics, Aligarh Muslim University, Aligarh-202002, India shakir50@ 2000 Mathematics Subject Classification. 16W25, 16N60, 16U80 Throughout the discussion, unless otherwise mentioned, R denotes an asso- ciative ring(may be without unity)with center Z(R). An additive mapping F : R ?→ R is called a generalized derivation on R if there exists a deriva- tion d : R ?→ R such that F (xy) = F (x)y+xd(y), holds for all x, y ∈ R. An additive mapping G : R ?→ R is called a generalized left derivation on R if there exists a left derivation δ : R ?→ R such that G(xy) = xG(y)+ yδ(x), holds for all x, y ∈ R. In this article, we introduce the notion of generalized left derivations and obtained some recent results on generalized derivations to generalized left derivation. The main result state as follows: Theorem. Let R be a prime ring and I a non-zero ideal of R. Then the following conditions are equivalent: (i) If R admits a generalized left derivation G associated with a non-zero left derivation δ such that G(xy) ? xy ∈ Z(R) or G(xy) + xy ∈ Z(R) for all x, y ∈ I (ii) If R admits a generalized left derivation G associated with a non-zero left derivation δ such that G(xy) ? yx ∈ Z(R) or G(xy) + yx ∈ Z(R) for all x, y ∈ I (iii) R is commutative. In addition, we also established some related results and discuss some examples which demonstrates that R to be prime is essential in the hypoth- esis of our result. ICM 2006 – Madrid, 22-30 August 2006 1 ICM 2006 – Posters. Abstracts. Section 02 On universal central extensions of precrossed and crossed modules Daniel Arias*, Jose? Manuel Casas, Manuel Ladra Departamento de Matema?ticas, Universidad de Leo?n, Campus de Vegazana, Leo?n, E-24071, Spain demdam@unileon.es 2000 Mathematics Subject Classification. 20J05, 18G50 The notion of universal extension was introduced by Kervaire in [