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Jeno Szigeti

Publications and source records attributed to Jeno Szigeti.

At least 19 recordsLinked to original sources
arXiv2021

Lie properties in associative algebras

Let K be a field, then we exhibit two matrices in the full nxn matrix algebra M_{n}(K) which generate M_{n}(K) as a Lie K-algebra with the commutator Lie product. We also study Lie centralizers of a not necessarily commutative unitary algebra and obtain results which we hope will eventually be a step in the direction of, firstly, proving that a Lie-nilpotent K-subspace (or a sub Lie K-algebra) of a finite-dimensional associative algebra over K of index k (say) generates a Lie-nilpotent associative subalgebra of much higher nilpotency index, and secondly, in the light of the sharp upper bound for the maximum (K-)dimension of a Lie-nilpotent K-subalgebra of M_{n}(K) of index k (obtained earlier), finding an upper bound for the maximum dimension of a Lie-nilpotent (of index k) sub Lie K-algebra of M_{n}(K). Finally, the constructive elementary proof of the Skolem-Noether theorem for the matrix algebra M_{n}(K) (appeared in the American Math. Monthly), in conjunction with the well-known characteization of Lie automorphisms of M_{n}(K) (if the characteristic of K is different from 2 and 3) in terms of, amongst others, automorphisms and anti-automorhisms of M_{n}(K), leads us to a unifying approach to constructively describe automorphisms and anti-automorphisms of M_{n}(K).

math.RA↗
arXiv2019

A power Cayley-Hamilton identity for nxn matrices over a Lie nilpotent ring of index k

For an nxn matrix A over a Lie nilpotent ring R of index k, we prove that an invariant "power" Cayley-Hamilton identity of degree (n^2)2^{k-2} holds. The right coefficients are not uniquely determined by A, and the cosets lambda_i+D, with D the double commutator ideal R[[R,R],R]R of R, appear in the so-called second right characteristic polynomial of the natural image of A in the nxn matrix ring M_{n}(R/D) over the factor ring R/D.

math.RA↗
arXiv2018

Integrality over fixed rings of automorphisms in a Lie nilpotent setting

Let R be a Lie nilpotent algebra of index k over a field K of characteristic zero. If G is an n-element subgroup of Aut(R) of the K-automorphisms, then we prove that R is right integral over Fix(G) of degree n^k. In the presence of a primitive n-th root of unity e in K, for a K-automorphism d in Aut(R) with d^n=id, we prove that the skew polynomial algebra R[w,d] is right integral of degree n^k over Fix(d)[w^n].

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arXiv2015

Which self-maps appear as lattice endomorphisms?

Let f be a self-map of the set A. We give a necessary and sufficient condition for the existence of a lattice structure on A such that f becomes a lattice endomorphism with respect to this structure.

math.RA↗
arXiv2015

On Lie nilpotent rings and Cohen's Theorem

We study certain (two-sided) nil ideals and nilpotent ideals in a Lie nilpotent ring R. Our results lead us to showing that the prime radical rad(R) of R comprises the nilpotent elements of R, and that if L is a left ideal of R, then L+rad(R) is a two-sided ideal of R. This in turn leads to a Lie nilpotent version of Cohen's theorem, namely if R is a Lie nilpotent ring and every prime (two-sided) ideal of R is finitely generated as a left ideal, then every left ideal of R containing the prime radical of R is finitely generated (as a left ideal). For an arbitrary ring R with identity we also consider its so-called n-th Lie center Z_n(R), which is a Lie nilpotent ring of index n. We prove that if C is a commutative submonoid of the multiplicative monoid of R, then the subring of R generated by the union of Z_n(R) and C is also Lie nilpotent of index n.

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arXiv2014

A new class of supermatrix algebras defined by transitive matrices

We give a natural definition for the transitivity of a matrix. Using an endomorphism d of a base ring R and a transitive nxn matrix over the center Z(R), we construct the subalgebra M_{n}(R,d,T) of the full nxn matrix algebra M_{n}(R) consisting of the so called nxn supermatrices. If the n-th power of d is the identity and T satisfies some extra conditions, then we exhibit an embedding of R into M_{n}(R,d,T). An other result is that M_{n}(R,d,T) is closed with respect to taking the (pre)adjoint. If R is Lie nilpotent and A is in M_{n}(R,d,T), then the use of the preadjoint and the corresponding determinants and characteristic polynomials yields a Cayley-Hamilton identity for A with right coefficients in the fixed ring Fix(d). The presence of a primitive n-th root of unity and the condition that the n-th power of d is the identity guarantee the right integrality of a Lie nilpotent R over Fix(d). We present essentially new supermatrix algebras over the Grassmann algebra.

math.RA↗
arXiv2009

Algebras generated by two quadratic elements

Let K be a field of any characteristic and let R be an algebra generated by two elements satisfying quadratic equations. Then R is a homomorphic image of F=K for suitable a,b,c,d in K. We establish that F can be embedded into the 2x2 matrix algebra M_2(E[t]) with entries from the polynomial algebra E[t] over the algebraic closure E of K and that F and M_2(E) satisfy the same polynomial identities as K-algebras. When the quadratic equations have double zeros, our result is a partial case of more general results by Ufnarovskij, Borisenko and Belov from the 1980's. When each of the equations has different zeros, we improve a result of Weiss, also from the 1980's.

math.RA↗
arXiv2009

Centralizers in endomorphism rings

We prove that the centralizer Cen(f) in Hom_R(M,M) of a nilpotent endomorphism f of a finitely generated semisimple left R-module M (over an arbitrary ring R) is the homomorphic image of the opposite of a certain Z(R)-subalgebra of the full m x m matrix algebra M_m(R[z]), where m is the dimension (composition length) of ker(f). If R is a local ring, then we provide an explicit description of the above Cen(f). If in addition Z(R) is a field and R/J(R) is finite dimensional over Z(R), then we give a formula for the Z(R)-dimension of Cen(f). If R is a local ring, f is as above and g is an arbitrary element of Hom_R(M,M), then we give a complete description of the containment Cen(f) in Cen(g) in terms of an appropriate R-generating set of M. Using our results about nilpotent endomorphisms, for an arbitrary (not necessarily nilpotent) linear map f in Hom_K(V,V) of a finite dimensional vector space V over a field K we determine the PI-degree of Cen(f) and give other information about the polynomial identities of Cen(f).

math.RA↗
arXiv2007

Subrings which are closed with respect to taking the inverse

Let S be a subring of the ring R. We investigate the question of whether S intersected by U(R) is equal to U(S) holds for the units. In many situations our answer is positive. There is a special emphasis on the case when R is a full matrix ring and S is a structural subring of R defined by a reflexive and transitive relation.

math.RA↗
arXiv2007

Linear algebra in lattices, the Fitting lemma

Lattice theoretical generalizations of some classical linear algebra results are formulated. A vector space is replaced by its subspace lattice and a linear map is replaced by the induced lattice map. This map is a complete join homomorphism and satisfies some additional natural conditions. The object of our study is a pair consisting of a complete lattice and one of its complete join endomorphisms. We prove the Fitting lemma for such a pair.

math.RA↗
arXiv2007

A half-space approach to order dimension

The aim of the present paper is to investigate the half-spaces in the convexity structure of all quasiorders on a given set and to use them in an alternative approach to classical order dimension. The main result states that linear orders can almost always be replaced by half-space quasiorders in the definition of the dimension of a partially ordered set.

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