Lie rings satisfying the Engel condition

1. Let be a Lie ring in which the product of elements x and y is denoted by xy. The inner derivations of , i.e. the mappings X:a→ax for fixed elements x of , form a Lie ring under the product [X, Y] = XY – YX, and the mapping x→ X is a homo-morphism of onto . We shall say that satisfies the nth Engel condition if Xn = 0 for all X in , i.e. iffor all a, x; in . If satisfies the maximum condition on subrings, it is known (1) that this condition implies the nilpotence of ; indeed, must then be nilpotent even if the integer n is allowed to depend on the element X of . We consider here the case in which n is independent of X but does not necessarily satisfy the maximum condition, and inquire whether is then nilpotent.

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A problem in lie rings

Glasgow Mathematical Journal ◽

10.1017/s0017089500004286 ◽

1980 ◽

Vol 21 (2) ◽

pp. 139-142

Author(s):

James Wiegold

Keyword(s):

Lie Algebra ◽

Finite Groups ◽

Fundamental Theorem ◽

Characteristic P ◽

Lie Rings ◽

Locally Nilpotent ◽

Engel Condition ◽

Prime Exponent

An important step in the proof of Kostrikin's fundamental theorem [2] on finite groups of prime exponent is the following result.Theorem 1. Let L be a Lie algebra of characteristic p satisfying the t-th Engel condition for some t < p, and suppose that L is generated by elements that are right-Engel of length 2. Then L is locally nilpotent.

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Non-associative Gröbner bases, finitely-presented Lie rings and the Engel condition, II

Journal of Symbolic Computation ◽

10.1016/j.jsc.2008.04.007 ◽

2009 ◽

Vol 44 (7) ◽

pp. 786-800 ◽

Cited By ~ 3

Author(s):

Serena Cicalò ◽

Willem A. de Graaf

Keyword(s):

Gröbner Bases ◽

Grobner Bases ◽

Lie Rings ◽

Engel Condition ◽

Finitely Presented

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A problem in lie rings

Glasgow Mathematical Journal ◽

10.1017/s0017089500004109 ◽

1980 ◽

Vol 21 (1) ◽

pp. 139-142

Author(s):

James Wiegold

Keyword(s):

Lie Algebra ◽

Finite Groups ◽

Fundamental Theorem ◽

Characteristic P ◽

Lie Rings ◽

Locally Nilpotent ◽

Engel Condition ◽

Prime Exponent

An important step in the proof of Kostrikin's fundamental theorem [2] on finite groups of prime exponent is the following result.Theorem 1. Let L be a Lie algebra of characteristic p satisfying the t-th Engel condition for some t<p, and suppose that L is generated by elements that are right-Engel of length 2. Then L is locally nilpotent.

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Lie rings and the Engel condition

Journal of Algebra ◽

10.1016/0021-8693(74)90070-2 ◽

1974 ◽

Vol 31 (2) ◽

pp. 287-292 ◽

Cited By ~ 6

Author(s):

Amiram Braun

Keyword(s):

Lie Rings ◽

Engel Condition

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Derivations of differentially semiprime rings

Asian-European Journal of Mathematics ◽

10.1142/s1793557119500797 ◽

2019 ◽

Vol 12 (05) ◽

pp. 1950079

Author(s):

Ahmad Al Khalaf ◽

Iman Taha ◽

Orest D. Artemovych ◽

Abdullah Aljouiiee

Keyword(s):

Commutative Ring ◽

Semiprime Ring ◽

Prime Rings ◽

Commutative Case ◽

Lie Ring ◽

Lie Rings ◽

Semiprime Rings ◽

Torsion Free

Earlier D. A. Jordan, C. R. Jordan and D. S. Passman have investigated the properties of Lie rings Der [Formula: see text] of derivations in a commutative differentially prime rings [Formula: see text]. We study Lie rings Der [Formula: see text] in the non-commutative case and prove that if [Formula: see text] is a [Formula: see text]-torsion-free [Formula: see text]-semiprime ring, then [Formula: see text] is a semiprime Lie ring or [Formula: see text] is a commutative ring.

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