Extended Jacobson Density Theorem for Rings with Derivations
by
Kostia Beidar
National Cheng Kung University, Tainan, Taiwan 701
Coauthors: M. Brešar

In what follows A is a ring with simple left module M and D=End(_AM). Given an element a in A, we denote by L_a the linear transformation of M_D defined by left multiplication by a. A derivation d:A --> A is said to be M-inner if there exists an additive map T:M --> M such that L_a^d=[L_a, T]. If d is not M-inner, then it is called an M-outer derivation. If A is a primitive ring and M is a faithful module, then an X-inner derivation (in sense of Kharchenko) is M-inner, but the converse is no longer true. Moreover one and the same derivation may be M-inner and N-outer for some other (faithful) simple A-module N. We denote by Der(A) the Lie ring of all derivations of A and by D_M(A) the Lie subring of all M-inner derivations of A. Elements d₁, d₂, ... , d_n are said to be linearly dependent over D modulo D_M(A) if there exist \alpha₁, \alpha₂, ... , \alpha_n in D not all 0 and \delta in D_M(A) such that \sum_i=1ⁿa^d_ix\alpha_i+a^dx=0 for all a in A and x in M.

Recently Bresar and Semrl proved that if A is a left primitive algebra algebra over a field F with faithful simple left module M such that F=End(_AM) and d:A --> A is an M-outer derivation of the algebra A, then for any linearly independent over F elements x₁, x₂, ... , x_n in M and any y₁, z₁, ... , y_n, z_n in M there exists a in A such that ax_i=y_i and a^dx_i=z_i. The Jacobson density theorem has been generalized in various directions by Amitsur, Fuller, Johnson and Wong, Kezlan, Koh and Newborn, Rowen, Zelmanowitz and others. The following main result of the talk is a generalization of both the Jacobson density theorem and the result of Bresar and Semrl.

Theorem Let d₁, d₂, ... , d_n in Der(A) be linearly independent over D modulo D_M(A), let m₁, m₂, ... , m_n be positive integers such that if char(D)=p > 0, then each m_i < p. Let \Omega be the set of all compositions of the form d₁^k₁d₂^k₂ ... d_n^k_n where each 0 <= k_i <= m_i (if each k_i=0, then d₁^k₁d₂^k₂ ... d_n^k_n=id_A). Let x₁, x₂, ... , x_r in M be linearly independent over D and let y_i\omega in M, 1 <= i <= r, \omega in \Omega. Then there exists a in A such that a^\omega x_i=y_i\omega for all i=1, 2, ... , r and \omega in \Omega.

The proof of the theorem is based on the construction of a module M_\Omega determined by M and \Omega. The theorem is equivalent to the fact that M_\Omega is a local module. Thus to prove the theorem it is enough to show that M_\Omega is local. We also present some applications of the theorem to the study of derivations in Ring Theory and Banach Algebras Theory.

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M. Bresar, P. Semrl, On locally linearly dependent operators and derivations, Trans. Amer. Math. Soc., to appear.

Date received: December 1, 1998

Copyright © 1998 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cabw-35.