Lie solvable rings

Abstract

Leti?(/<) denote the associated Lie ring of an associative ring R with identity 1^0 under the Lie multiplication [x, y] = xv-vx with x, y e R. Further, suppose that the Lie ringi?(R) is solvable of length n. It has been proved that if 3 is invertible in R, then the ideal J of R generated by all elements [[[*,, jc2],[x3, x4]], xs], x¡, x2, x3, x4, x¡ e R, is nilpotent of index at most I (19 ■ 10"~3-1) for n > 3. Also, if 2 and 3 are both invertible in R, then the ideal I of R generated by all elements [x, y), x, y s R, is a nil ideal of R. Some applications to Lie solvable group rings are also given.

On the Jacobson radical and unit groups of group algebras

“…If further 2 and 3 are invertible in R and the associated Lie ring L(R) is solvable, then γ 2 (L(R))R = δ 1 (L(R))R is a nil ideal of R by Sharma and Srivastava [7,Theorem 2.4]. Since nil ideals are always contained in the Jacobson radical, we have, in this situation, γ 2 (L(R))R ⊆ J(R) and thus R/J(R) is commutative.…”

Associative rings with metabelian adjoint group

“…For instance, it is well known that the ring R of all (2 × 2)-matrices over a commutative radical domain of characteristic 2 is radical and satisfies the equality [δ 2 (R), R] = 0, whereas the adjoint group of R contains a non-abelian free subgroup and so is non-soluble. Later Zalesskii and Smirnov [13] and independently Sharma and Srivastava [9] have shown that in every Lie soluble ring R the ideal generated by [δ 2 (R), R] is nilpotent.…”

Radical Rings with Soluble Adjoint Groups

“…For instance, it was proved by Zalesskii and Smirnov [12] and independently by Sharma and Srivastava [9] that every Lie-soluble ring R has a nilpotent ideal I whose factor ring R/I is centre-by-metabelian as a Lie ring. However, the adjoint group R…”