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A symbolic Lie p -ring presentation defines a family of nilpotent Lie rings with p n elements for infinitely many primes p and a fixed positive integer n. Symbolic Lie p -ring presentations are used in the classification of isomorphism types of nilpotent Lie rings of order p n for all primes p and n ≤ 7. We describe an algorithm to compute the Schur multipliers of all nilpotent Lie rings in the family defined by a symbolic Lie p -ring presentation. We apply this to determine the Schur multipliers of all nilpotent Lie rings of order dividing p 6 for all primes p ≥ 5. Via the Lazard correspondence this yields the Schur multipliers of all groups of order dividing p 6 for all primes p ≥ 5. [ABSTRACT FROM AUTHOR]

In the present paper, we prove that every 2-local inner derivation on the Lie ring of skew-symmetric matrices over a commutative ring is an inner derivation. We also apply our technique to various Lie algebras of infinite-dimensional skew-adjoint matrix-valued maps on a set and prove that every 2-local spatial derivation on such algebras is a spatial derivation. A similar technique is applied to the same Lie algebras and proved that every local spatial derivation on such algebras is a spatial derivation. [ABSTRACT FROM AUTHOR]

Let ℳ be a structure of a signature Σ. For any ordered tuple a ¯ = a 1 ... a n of elements of ℳ, tp M a ¯ denotes the set of formulas θ(x1, ..., xn) of a first-order language over Σ with free variables x1,... , xn such that M = θ a 1 ... a n . A structure ℳ is said to be strongly ω-homogeneous if, for any finite ordered tuples a ¯ and b ¯ of elements of ℳ, the coincidence of tp M a ¯ and tp M b ¯ implies that these tuples are mapped into each other (componentwise) by some automorphism of the structure ℳ. A structure ℳ is said to be prime in its theory if it is elementarily embedded in every structure of the theory Th (ℳ). It is proved that the integral group rings of finitely generated relatively free orderable groups are prime in their theories, and that this property is shared by the following finitely generated countable structures: free nilpotent associative rings and algebras, free nilpotent rings and Lie algebras. It is also shown that finitely generated non-Abelian free nilpotent associative algebras and finitely generated non-Abelian free nilpotent Lie algebras over uncountable fields are strongly ω-homogeneous. [ABSTRACT FROM AUTHOR]

Earlier properties of Lie rings DerR of derivations in commutative differentially prime rings R was investigated by many authors. We study Lie rings DerR in the non-commutative case and shown that if R is a D-prime ring of characteristic ≠2, then D is a prime Lie ring or R is a commutative ring. [ABSTRACT FROM AUTHOR]

Let be an associative ring. We characterize rings with simple Lie ring of all Lie derivations, reduced noncommutative Noetherian ring with the simple Lie ring of all derivations and obtain some properties of -torsion-free rings with the simple Lie ring of all Jordan derivations. [ABSTRACT FROM AUTHOR]

We investigate properties of FC-rings (i.e. rings R in which the centralizer CR (a) of any element a ∊ R is of finite index in R)and, in particular, characterize left Artinian rings with a finite set of all derivations Der R (respectively inner derivations Der R (respectively inner derivations IDer R).We show that if R is a Jacobson radical ring in which its adjoint group R° has a finite number of conjugacy classes, then R=Rp1 ⊕....⊕ Rpt ⊕ D is a ring direct sum of Jacobson radical rings Rpi and D, where the additive group D+ is a torsion-free divisible group, the adjoint group D° is a group with a finite number of conjugacy classes, R+pi is a finite pi- group (i=l,....,t) and p1,....,pt are pairwise distinct primes. [ABSTRACT FROM AUTHOR]

Automorphic loops are loops in which all inner mappings are automorphisms. This variety of loops includes, for instance, groups and commutative Moufang loops. We study uniquely 2-divisible automorphic loops, particularly automorphic loops of odd order, from the point of view of the associated Bruck loops (motivated by Glauberman's work on uniquely 2-divisible Moufang loops) and the associated Lie rings (motivated by a construction of Wright). We prove that every automorphic loop Q of odd order is solvable and contains an element of order p for every prime p dividing |Q|, and that |S| divides |Q| for every subloop S of Q. There are no finite simple nonassociative commutative automorphic loops, and there are no finite simple nonassociative automorphic loops of order less than 2500. We show that if Q is a finite simple nonassociative automorphic loop, then the socle of the multiplication group of Q is not regular. The existence of a finite simple nonassociative automorphic loop remains open. Let p be an odd prime. Automorphic loops of order p or p² are groups, but there exist nonassociative automorphic loops of order p³, some with trivial nucleus (center) and of exponent p. We construct nonassociative "dihedral" automorphic loops of order 2n for every n > 2, and show that there are precisely p-2 nonassociative automorphic loops of order 2p, all of them dihedral. [ABSTRACT FROM AUTHOR]

We identify the spaces of homogeneous polynomials in two variables 𝕂[Yk,XYk−1, ⋅,Xk] among representations of the Lie ring. This amounts to constructing a compatible 𝕂-linear structure on some abstract-modules, whereis viewed as a Lie ring. [ABSTRACT FROM AUTHOR]

Suppose that a finite p-group P admits a Frobenius group of automorphisms FH with kernel F that is a cyclic p-group and with complement H. It is proved that if the fixed-point subgroup Cp(H) of the complement is nilpotent of class c, then P has a characteristic subgroup of index bounded in terms of c, |Cp(F)|, and |F| whose nilpotency class is bounded in terms of c and |H | only. Examples show that the condition of F being cyclic is essential. The proof is based on a Lie ring method and a theorem of the authors and P. Shumyatsky about Lie rings with a metacyclic Frobenius group of automorphisms FH. It is also proved that P has a characteristic subgroup of (|Cp(F)|, |F|)-bounded index whose order and rank are bounded in terms of |H| and the order and rank of Cp(H), respectively, and whose exponent is bounded in terms of the exponent of Cp(H). [ABSTRACT FROM AUTHOR]

LetRbe a 2-torsion free commutative ring with involution, and δ a nonzero derivation ofR. LetSbe the set of symmetric elements inR, and letKbe the set of anti-symmetric elements inR. In this article, we investigate the semiprimeness of the Lie ringsSδ when δ is symmetric andKδ when δ is anti-symmetric. [ABSTRACT FROM AUTHOR]

Suppose that a finite group G admits a Frobenius group of automorphisms with kernel F and complement H such that the fixed-point subgroup of F is trivial: . In this situation various properties of G are shown to be close to the corresponding properties of . By using Clifford's theorem it is proved that the order is bounded in terms of and , the rank of G is bounded in terms of and the rank of , and that G is nilpotent if is nilpotent. Lie ring methods are used for bounding the exponent and the nilpotency class of G in the case of metacyclic . The exponent of G is bounded in terms of and the exponent of by using Lazard's Lie algebra associated with the Jennings-Zassenhaus filtration and its connection with powerful subgroups. The nilpotency class of G is bounded in terms of and the nilpotency class of by considering Lie rings with a finite cyclic grading satisfying a certain `selective nilpotency' condition. The latter technique also yields similar results bounding the nilpotency class of Lie rings and algebras with a metacyclic Frobenius group of automorphisms, with corollaries for connected Lie groups and torsion-free locally nilpotent groups with such groups of automorphisms. Examples show that such nilpotency results are no longer true for non-metacyclic Frobenius groups of automorphisms. [ABSTRACT FROM AUTHOR]

Let R be a ring. If we replace the original associative product of R with their canonic Lie product, or [a, b] = ab - ba for every a, b in R, then R would be a Lie ring. With this new product the additive commutator subgroup of R or [R, R] is a Lie subring of R. Herstein has shown that in a simple ring R with characteristic unequal to 2, any Lie ideal of R either is contained in Z(R), the center of R or contains [R, R]. He also showed that in this situation the Lie ring [R, R]/Z[R, R] is simple. We give an alternative matrix proof of these results for the special case of simple artinian rings and show that in this case the characteristic condition can be more restricted. [ABSTRACT FROM AUTHOR]

Abstract: Suppose that a finite group G admits a Frobenius group of automorphisms FH of coprime order with cyclic kernel F and complement H such that the fixed-point subgroup of the complement is nilpotent of class c. It is proved that G has a nilpotent characteristic subgroup of index bounded in terms of c, , and whose nilpotency class is bounded in terms of c and only. This generalizes the previous theorem of the authors and P. Shumyatsky, where for the case of the whole group was proved to be nilpotent of -bounded class. Examples show that the condition of F being cyclic is essential. Results based on the classification provide reduction to soluble groups. Then representation theory arguments are used to bound the index of the Fitting subgroup. Lie ring methods are used for nilpotent groups. A similar theorem on Lie rings with a metacyclic Frobenius group of automorphisms FH is also proved. [Copyright &y& Elsevier]

An answer is obtained to I. V. Arzhantsev's question on definability of the structure of a semisimple Lie algebra by the multiplicative groupoid of L. [ABSTRACT FROM AUTHOR]

We discuss a method for classifying integrable equations on quad-graphs based on algebraic ideas. We assign a Lie ring to the equation and study the function describing the dimensions of linear spaces spanned by multiple commutators of the ring generators. This function grows exponentially in the general case. Examples show that it grows more slowly for integrable equations. We propose a classification scheme based on this observation. [ABSTRACT FROM AUTHOR]

In a recent study of Engel Lie rings, Serena Cicalo and Willem de Graaf have given a practical set of conditions for an additively finitely generated Lie ring L to satisfy an Engel condition. We present a simpler and more direct proof of this fact. Our main result generalizes this in the language of tensor algebra, and describes a relatively small generating set for the module generated by all n-th tensor powers of elements of a finitely generated -module M, in terms of a generating set for M. [ABSTRACT FROM AUTHOR]

Derivations of the ring of all finitary niltriangular matrices over an arbitrary associative ring with identity for any chain of matrix indices are described. Every Lie or Jordan derivation is a derivation of this ring modulo third hypercenter. [ABSTRACT FROM AUTHOR]

Let R be a 2-torsion free commutative ring with involution *, and δ a non-zero derivation of R. Let S be the set of symmetric elements in R and K the set of anti-symmetric elements in R. In this article, we investigate the simplicity and primeness of the Lie rings Sδ and Kδ when δ is a symmetric or anti-symmetric derivation. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring, and D a Lie subring and an R-submodule of Der(R) such that R is D-semiprime (or D-prime). We investigate the structure of the ideals of D as Lie rings. As a consequence, we give a necessary and sufficient condition for the ideals of D to be semiprime (or prime, respectively) Lie rings. [ABSTRACT FROM AUTHOR]

Abstract: Let A be an arbitrary (not necessarily associative or commutative) algebra over a field K. It is proved that if A has an ideal of finite codimension r satisfying a multilinear identity , then A also has an ideal satisfying the same identity that is invariant under all automorphisms of A and has finite codimension bounded in terms of r and f. The result is stronger in characteristic zero, where f need not be multilinear. As a corollary, it is proved that if a locally nilpotent torsion-free group G has a normal subgroup H satisfying a multilinear commutator identity with quotient of finite rank r, then G also has a characteristic subgroup C satisfying the same identity with quotient of finite rank bounded in terms of r and ϰ. An example shows that the main result cannot be extended to algebras not over fields, even to Lie algebras over integers. An analogous example shows that the result on characteristic nilpotent subgroups with quotients of finite rank, which was proved by the authors earlier in torsion-free and periodic cases, cannot be extended to mixed nilpotent groups. [Copyright &y& Elsevier]

Abstract: Let p be a prime number. A finite nilpotent Lie ring of characteristic a power of p is called finite-p. A pro-p Lie ring is an inverse limit of finite-p Lie rings. Pro-p Lie rings play a role in Lie theory similar to that played by pro-p groups in group theory. Every pro-p Lie ring admits the structure of a Lie algebra over the p-adic integers; furthermore, every p-adic Lie algebra of finite rank as a p-adic module has an open pro-p subalgebra. We make a detailed study of pro-p Lie rings in terms of various properties, including their topology, Prüfer rank, subring growth, and p-adic module structure. In particular, we prove the equivalence of the following conditions for a finitely generated pro-p Lie ring L: L has finite Prüfer rank; L is isomorphic to a closed subring of for some p-adic module V of finite rank; and, for sufficiently large n, the Lie -subalgebra is not an open section of L. By reducing to the pro-p Lie ring case, we also prove that all Engelian pro-finite Lie rings are locally nilpotent. This is a Lie theoretic analogue of Zelmanov''s theorem which states that every periodic pro-p group is locally finite. [Copyright &y& Elsevier]

Let R be a 2-torsionfree commutative ring, and D a Lie subring and an R-submodule of Der(R) such that R is D-prime. Then any ideal of the Lie ring D is itself a prime Lie ring. [ABSTRACT FROM AUTHOR]

Let R be a 2-torsion free commutative ring with identity, and δ a nonzero derivation of R such that R is δ-prime. Then Rδ is a prime Lie ring and any nonzero ideal of Rδ contains an ideal of the form Jδ where J is a nonzero δ-ideal of R. [ABSTRACT FROM AUTHOR]

P. Shumyatsky’s question 11.126 in the “Kourovka Notebook” is answered in the affirmative: it is proved that there exist a constant c and a function of a positive integer argument f(m) such that if a finite group G admits an automorphism ϕ of order 4 having exactly m fixed points, then G has a normal series G ⩾ H ⩽ N such that |G/H| ⩽ f(m), the quotient group H/N is nilpotent of class ⩽ 2, and the subgroup N is nilpotent of class ⩽ c (Thm. 1). As a corollary we show that if a locally finite group G contains an element of order 4 with finite centralizer of order m, then G has the same kind of a series as in Theorem 1. Theorem 1 generalizes Kovács’ theorem on locally finite groups with a regular automorphism of order 4, whereby such groups are center-by-metabelian. Earlier, the first author proved that a finite 2-group with an almost regular automorphism of order 4 is almost center-by-metabelian. The proof of Theorem 1 is based on the authors’ previous works dealing in Lie rings with an almost regular automorphism of order 4. Reduction to nilpotent groups is carried out by using Hall-Higman type theorems. The proof also uses Theorem 2, which is of independent interest, stating that if a finite group S contains a nilpotent subgroup T of class c and index |S: T | = n, then S contains also a characteristic nilpotent subgroup of class ⩽ c whose index is bounded in terms of n and c. Previously, such an assertion has been known for Abelian subgroups, that is, for c = 1. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring, G a group and RG its group ring. Let φ: RG → RG denote the R-linear extension of an involution φ defined on G. An element x in RG is said to be antisymmetric if φ(x) = -x. A characterization is given of when the antisymmetric elements $(RG)^{-}_{\varphi}$ of RG commute except when Char(R) = 3. [ABSTRACT FROM AUTHOR]

From a finite oriented graph G, finite-dimensional graded nilpotent Lie rings(G) and(G) are naturally constructed; these rings are related to subtrees and connected subgraphs of G, respectively. Diverse versions of these constructions are also suggested. Moreover, an embedding of Lie rings of the form(G) in the adjoint Lie rings of finite-dimensional associative rings (also determined by the graph G) is indicated. [ABSTRACT FROM AUTHOR]