Multilinear Components of the Prime Subvarieties of the Variety Var(M2(F))

We describe the multilinear components of the prime subvarieties of the variety Var(M ₂(F)) generated by the matrix algebra of order 2 over a field of characteristic p>0.     

Remarks on the prime varieties

We study the prime varieties of associative algebras over infinite fields of characteristicp. We prove a few properties of the multilinear components ofT-primeT-ideals and describe the prime subvarieties of the variety of the algebras satisfying all the identities of the algebraM ₂(F) and the identityx ^p =0.     

On Some Extremal Varieties of Associative Algebras

Suppose that F is a field of prime characteristic p and V _p is the variety of associative algebras over F defined by the identities [[x, y], z] = 0 and x ^p = 0 if p > 2 and by the identities [[x, y], z] = 0 and x ⁴ = 0 if p = 2 (here [x, y] = xy ? yx). As is known, the free algebras of countable rank of the varieties V _p contain non-finitely generated T-spaces. We prove that the varieties V _p are minimal with respect to this property.     

Identities of symmetric and skew-symmetric matrices in characteristicp

It is well known how the Kostant-Rowen Theorem extends the validity of the famous Amitsur-Levitzki identity to skew-symmetric matrices. Here we give a general method, based on a graph theoretic approach, for deriving extensions of known permanental-type identities to skew-symmetric and symmetric matrices over a commutative ring of prime characteristic. Our main result has a typical Kostant-Rowen flavour: IfM≥p[n+1/2] then $C_M (X,Y) = \sum\limits_{\alpha ,\beta \in Sym(M)} {x_{\alpha (1)} y_{\beta (1)} x_{\alpha (2)} y_{\beta (2)} } ...x_{\alpha (M)} y_{\beta (M)} = 0$ is an identity onM _n ^? (Ω), the set ofnxn skew-symmetric matrices over a commutative ring Ω withp1_Ω=0 (provided that $P > \sqrt {[n + 1/2)} $ ). Otherwise, the stronger conditionM≥pn implies thatC _M(X,Y)=0 is an identity on the full matrix ringM _n(Ω).     

n-almost prime submodules

Let R be a commutative ring with identity. A proper submodule N of an R-module M will be called prime [resp. n-almost prime], if for r ∈ R and a ∈ M with ra ∈ N [resp. ra ∈ N \ (N: M) ^n?1 N], either a ∈ N or r ∈ (N: M). In this note we will study the relations between prime, primary and n-almost prime submodules. Among other results it is proved that:

1. If N is an n-almost prime submodule of an R-module M, then N is prime or N = (N: M)N, in case M is finitely generated semisimple, or M is torsion-free with dim R = 1.
2. Every n-almost prime submodule of a torsion-free Noetherian module is primary.
3. Every n-almost prime submodule of a finitely generated torsion-free module over a Dedekind domain is prime.
4. There exists a finitely generated faithful R-module M such that every proper submodule of M is n-almost prime, if and only if R is Von Neumann regular or R is a local ring with the maximal ideal m such that m ² = 0.
5. If I is an n-almost prime ideal of R and F is a flat R-module with IF ≠ F, then IF is an n-almost prime submodule of F.

     

Subvarieties of the matrix variety of second order

LetR _s be the subalgebra ofM ₂(K[t]/(t ^s )) generated bye ₁₁,e ₂₂,te ₁₂ andte ₂₁, whereK is a field of characteristic 0,K[t] is the polynomial algebra in one variablet and (t ^s ) is the principal ideal inK[t], generated byt ^s . The main result of this paper is that we have described theT-idealT(R _s ). Besides the two matrix polynomial identities — the standart identityS ₄ and the identity of Hall, thisT-ideal is generated by one more explicitly given identity. The algebrasR _s are interesting due to the fact that the proper identities of any subvarietyu of the variety ?=varM ₂(K), generated by the matrix algebraM ₂(K) of second order overK, asymptoticaly coincide with the proper identities of someR _s .     

Markov-Bernstein type inequalities for constrained polynomials with real versus complex coefficients

LetP _n,k ^c denote the set of all polynomials of degree at mostn withcomplex coefficients and with at mostk(0≤k≤n) zeros in the open unit disk. Let denote the set denote the set of all polynomials of degree at mostn withreal coefficients and with at mostk(0≤k≤n) zeros in the open unit disk. Associated with0≤k≤n andx∈[?1, 1], let $B_{n,k,x}^* : = \max \{ \sqrt {\frac{{n(k + 1)}}{{1 - x^2 }}} ,n\log (\frac{e}{{1 - x^2 }}\} ,B_{n,k,x}^* : = \sqrt {\frac{{n(k + 1)}}{{1 - x^2 }}} ,$ , andM _n,k ^* ?max{n(k+1),nlogn},M _n,k ?n(k+1). It is shown that $M_{n,k}^* : = \max \{ n(k + 1),n\log n\} ,M_{n,k}^* :n(k + 1)$ for everyx∈[?1, 1], wherec ₁>0 andc ₂>0 are absolute constants. Here ‖·‖_[?1,1] denotes the supremum norm on [?1,1]. This result should be compared with the inequalities $c3\min \{ B_{n,k,x,} B_{n,,k,} \} \leqslant _{p \in P_{n,k} }^{\sup } \frac{{|p'(x)|}}{{||p||[1,1]}} \leqslant \{ B_{n,k,x,} B_{n,,k,} \} ,$ , for everyx∈[?1,1], wherec ₃>0 andc ₄>0 are absolute constants. The upper bound of this second result is also fairly recent; and it may be surprising that there is a significant difference between the real and complex cases as far as Markov-Bernstein type inequalities are concerned. The lower bound of the second result is proved in this paper. It is the final piece in a long series of papers on this topic by a number of authors starting with Erdös in 1940.     

Correlation of Poisson algebras and Lie algebras in the language of identities

In the paper, the varieties of Poisson algebras whose ideals of identities contain the identity {x, y}· {z, t} = 0 are studied, and the correlation of these varieties with varieties of Lie algebras is investigated. A variety of Poisson algebras of almost exponential growth is presented. An example of a variety of Poisson algebras with fractional exponent is also given.     

IDENTITIES FOR ALGEBRAS OBTAINED FROM THE CAYLEY-DICKSON PROCESS

The Cayley-Dickson process gives a recursive method of constructing a nonassociative algebra of dimension 2 ⁿ for all n ≥ 0, beginning with any ring of scalars. The algebras in this sequence are known to be flexible quadratic algebras; it follows that they are noncommutative Jordan algebras: they satisfy the flexible identity in degree 3 and the Jordan identity in degree 4. For the integral sedenion algebra (the double of the octonions) we determine a complete set of generators for the multilinear identities in degrees ≤ 5. Since these identities are satisfied by all flexible quadratic algebras, it follows that a multilinear identity of degree ≤ 5 is satisfied by all the algebras obtained from the Cayley-Dickson process if and only if it is satisfied by the sedenions.     

Asymptotics of the Codimensions <Emphasis Type="Italic">c</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> in the Algebra <Emphasis Type="Italic">F</Emphasis><Superscript>(7)</Superscript>

The paper studies the additive structure of the algebra F⁽⁷⁾, i.e., a relatively free associative countably generated algebra with the identity [x₁,..., x₇] = 0 over an infinite field of characteristic ≠ 2, 3. First, the space of proper multilinear polynomials in this algebra is investigated. As an application, estimates for the codimensions c_n = dimF_n⁽⁷⁾ are obtained, where F_n⁽⁷⁾ stands for the subspace of multilinear polynomials of degree n in the algebra F⁽⁷⁾.     

Cocharacters of Bilinear Mappings and Graded Matrices

Let M _k (F) be the algebra of k ×k matrices over a field F of characteristic 0. If G is any group, we endow M _k (F) with the elementary grading induced by the k-tuple (1,...,1,g) where g?∈?G, g ²?≠?1. Then the graded identities of M _k (F) depending only on variables of homogeneous degree g and g ^???1 are obtained by a natural translation of the identities of bilinear mappings (see Bahturin and Drensky, Linear Algebra Appl 369:95–112, 2003). Here we study such identities by means of the representation theory of the symmetric group. We act with two copies of the symmetric group on a space of multilinear graded polynomials of homogeneous degree g and g ^???1 and we find an explicit decomposition of the corresponding graded cocharacter into irreducibles.     

Limit T-spaces

Let F be a field of prime characteristic p and let V _p be the variety of associative algebras over F without unity defined by the identities [[x, y], z] = 0 and x ⁴ = 0 if p = 2 and by the identities [[x, y], z] = 0 and x ^p = 0 if p > 2 (here [x, y] = xy − yx). Let A/V _p be the free algebra of countable rank of the variety V _p and let S be the T-space in A/V _p generated by x ₁² x ₂² ⋯ x _k² + V ₂, where k ∈ ℕ if p = 2, and by {ie4170-01}, where k ∈ ℕ and α ₁, …, α _2k ∈ {0, p − 1} if p > 2. As is known, S is not finitely generated as a T-space. In the present paper, we prove that S is a limit T-space, i.e., a maximal nonfinitely generated T-space. As a corollary, we have constructed a limit T-space in the free associative F-algebra without unity of countable rank. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 1, pp. 135–159, 2007.     

Free zeropotent left distributive systems

We prove a result conjectured by J.Je[zcirc]ek in [9],namely that a zeropotent left self-distributive system need not be 3-trivial, i.e.,the identity x(yz)=0 does not follow from the identities x(yz)=(xy)(xz),xx=0 and x0=0x=0.The argument is a general scheme possibly working for various identities in the context of self-distributivity.     

Identities with Generalized Derivations in Prime Rings

In this paper we investigate identities with two generalized derivations in prime rings. We prove, for example, the following result. Let R be a prime ring of characteristic different from two and let F ₁, F ₂ : R → R be generalized derivations satisfying the relation F ₁(x)F ₂(x) + F ₂(x)F ₁(x) = 0 for all ${x \in R}$ . In this case either F ₁ = 0 or F ₂ = 0.     

A fully invariant ideal of alternative algebras

Let ϕ be an associative commutative ring with 1, containing 1/6, and A be an alternative ϕ-algebra. Let D be an associator ideal of A and H a fully invariant ideal of A, generated by all elements of the form h(y, z, t, x, x)=[{[y, z], t, x}-, x]+[{[y, x], z, x}-, t], where [x, y]=xy−yx, {x, y, z}-=[[x, y], z]−[[x, z], y]+2[x,[y, z]]. Here we consider an ideal Q=H∩D and prove that Q⁴=0 in the algebra A. If A is unmixed, then HD=0, DH=0, and Q²=0 in particular. If A is a finitely generated unmixed algebra, then the ideal H lies in its associative center and Q=0. It follows that any finitely generated purely alternative algebra satisfies the identity h(y,z,t,x,x)=0. We also show that a fully invariant ideal H₀ of the unmixed algebra A, generated by all elements of the form h(x, z, t, x, x), lies in its associative center and H₀∩D=0. Consequently, every purely alternative algebra satisfies the identity h(x,z,t,x,x)=0. Translated fromAlgebra i Logika, Vol. 36, No. 3, pp. 323–340, May–June, 1997.     

New Properties of the Lie Algebra Variety N2A

We continue to consider the properties of the almost polynomial growth variety of Lie algebras over a field of characteristic zero defined by the identity (x ₁ x ₂)(x ₃ x ₄)(x ₅ x ₆)?≡?0. Here we have constructed the bases of its multilinear parts and proved the formulas for the colength and codimension sequences of this variety.     

Commutator identities of the homotopes of (−1, 1)-algebras

We study the commutator algebras of the homotopes of (?1, 1)-algebras and prove that they are Malcev algebras satisfying the Filippov identity h _a (x, y, z) = 0 in the case of strictly (?1, 1)-algebras. We also proved that every Malcev algebra with the identities xy ³ = 0, xy ² z ² = 0, and h _a (x, y, z) = 0 is nilpotent of index at most 6.     

Polynomial Identities of M2,1(G)

We describe the multilinear identities of the superalgebra M _{2, 1}(G) of matrices over the Grassmann algebra, in the minimal possible degree, which is 9.