Polynomial identities of bicommutative algebras,Lie and Jordan elements

ABSTRACT

An algebra with identities a(bc)?=?b(ac), (ab)c?=?(ac)b is called bicommutative. We construct list of identities satisfied by commutator and anti-commutator products in a free bicommutative algebra. We give criterions for elements of a free bicommutative algebra to be Lie or Jordan.     

Bounding the Derived Length for a Given Set of Character Degrees

Let G be a finite solvable group with {1, a, b, c, ab, ac} as the character degree set, where a ,b, and c are pairwise coprime integers greater than 1. We show that the derived length of G is at most 4. This verifies that the Taketa inequality, dl(G) ≤ |cd(G)|, is valid for solvable groups with {1, a, b, c, ab, ac} as the character degree set. Also, as a corollary, we conclude that if a, b, c, and d are pairwise coprime integers greater than 1 and G is a solvable group such that cd(G) = {1, a, b, c, d, ac, ad, bc, bd}, then dl(G) ≤ 5. Finally, we construct a family of solvable groups whose derived lengths are 4 and character degree sets are in the form {1, p, b, pb, q ^p , pq ^p }, where p is a prime, q is a prime power of an odd prime, and b > 1 is integer such that p, q, and b are pairwise coprime. Hence, the bound 4 is the best bound for the derived length of solvable groups whose character degree set is in the form {1, a, b, c, ab, ac} for some pairwise coprime integers a, b, and c.     

Semiprimality and Nilpotency of Nonassociative Rings Satisfying x(yz) = y(zx)

In this article we study nonassociative rings satisfying the polynomial identity x(yz) = y(zx), which we call “cyclic rings.” We prove that every semiprime cyclic ring is associative and commutative and that every cyclic right-nilring is solvable. Moreover, we find sufficient conditions for the nilpotency of cyclic right-nilrings and apply these results to obtain sufficient conditions for the nilpotency of cyclic right-nilalgebras.     

[a,b]-factorization of a graph

Let a and b be integers such that 0 ? a ? b. Then a graph G is called an [a, b]-graph if a ? d_G(x) ? b for every x ? V(G), and an [a, b]-factor of a graph is defined to be its spanning subgraph F such that a ? d_F(x) ? b for every vertex x, where d_G(x) and d_F(x) denote the degrees of x in G and F, respectively. If the edges of a graph can be decomposed into [a.b]-factors then we say that the graph is [2a, 2a]-factorable. We prove the following two theorems: (i) a graph G is [2a, 2b)-factorable if and only if G is a [2am,2bm]-graph for some integer m, and (ii) every [8m + 2k, 10m + 2k]-graph is [1,2]-factorable.     

Algebras with bracket

An algebra with bracket is an associative algebra A equipped with a bilinear operation [−,−] satisfying [a · b, c] = [a, c]·b+a · [b, c]. Our main result claims that the operad corresponding to algebras with bracket is Koszul.     

On tortkara triple systems

The commutator [a,b] = ab?ba in a free Zinbiel algebra (dual Leibniz algebra) is an anticommutative operation which satisfies no new relations in arity 3. Dzhumadildaev discovered a relation T(a,b,c,d) which he called the tortkara identity and showed that it implies every relation satisfied by the Zinbiel commutator in arity 4. Kolesnikov constructed examples of anticommutative algebras satisfying T(a,b,c,d) which cannot be embedded into the commutator algebra of a Zinbiel algebra. We consider the tortkara triple product [a,b,c] = [[a,b],c] in a free Zinbiel algebra and use computer algebra to construct a relation TT(a,b,c,d,e) which implies every relation satisfied by [a,b,c] in arity 5. Thus, although tortkara algebras are defined by a cubic binary operad (with no Koszul dual), the corresponding triple systems are defined by a quadratic ternary operad (with a Koszul dual). We use computer algebra to construct a relation in arity 7 satisfied by [a,b,c] which does not follow from the relations of lower arity. It remains an open problem to determine whether there are further new identities in arity n≥9.     

On Composition series relative to a module

ABSTRACT

A commutative algebra with the identity (a * b) * (c * d) ? (a * d) * (c * b) = (a, b, c) * d ? (a, d, c) * b is called Novikov–Jordan. Example: K[x] under multiplication a * b = ?(ab) is Novikov–Jordan. A special identity for Novikov–Jordan algebras of degree 5 is constructed. Free Novikov–Jordan algebras with q generators are exceptional for any q ≥ 1.

     

Abelian and Hamiltonian groupoids

In this work, we investigate some groupoids that are Abelian algebras and Hamiltonian algebras. An algebra is Abelian if for every polynomial operation and for all elements a, b, [`(c)] \bar{c} , [`(d)] \bar{d} the implication t( a,[`(c)] ) = t( a,[`(d)] ) T t( b,[`(c)] ) = t( b,[`(d)] ) t\left( {a,\bar{c}} \right) = t\left( {a,\bar{d}} \right) \Rightarrow t\left( {b,\bar{c}} \right) = t\left( {b,\bar{d}} \right) holds. An algebra is Hamiltonian if every subalgebra is a block of some congruence on the algebra. R. J. Warne in 1994 described the structure of the Abelian semigroups. In this work, we describe the Abelian groupoids with identity, the Abelian finite quasigroups, and the Abelian semigroups S such that abS = aS and Sba = Sa for all a, b ∈ S. We prove that a finite Abelian quasigroup is a Hamiltonian algebra. We characterize the Hamiltonian groupoids with identity and semigroups under the condition of Abelianity of these algebras.     

On the semigroup nilpotency and the Lie nilpotency of associative algebras

To each associative ringR we can assign the adjoint Lie ringR ⁽⁻⁾ (with the operation(a,b)=ab−ba) and two semigroups, the multiplicative semigroupM(R) and the associated semigroupA(R) (with the operationaob=ab+a+b). It is clear that a Lie ringR ⁽⁻⁾ is commutative if and only if the semigroupM(R) (orA(R)) is commutative. In the present paper we try to generalize this observation to the case in whichR ⁽⁻⁾ is a nilpotent Lie ring. It is proved that ifR is an associative algebra with identity element over an infinite fieldF, then the algebraR ⁽⁻⁾ is nilpotent of lengthc if and only if the semigroupM(R) (orA(R)) is nilpotent of lengthc (in the sense of A. I. Mal'tsev or B. Neumann and T. Taylor). For the case in whichR is an algebra without identity element overF, this assertion remains valid forA(R), but fails forM(R). Another similar results are obtained. Translated fromMatematicheskie Zametki, Vol. 62, No. 4, pp. 510–519, October, 1997. Translated by A. I. Shtern