Jeno Szigeti

On the characteristic polynomial of supermatrices

We prove that the coefficients of the so-called right 2-characteristic polynomial of a supermatrix over a Grassmann algebraG=G0⊕G1 are in the even componentG0 ofG. As a consequence, we obtain that the algebra ofn×n supermatrices is integral of degreen2 overG0.

Which self-maps appear as lattice endomorphisms?

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

LINEAR ORDERS ON RINGS

The general ideas introduced in Radeleczki and Szigeti (2004) are adapted to investigate quasi cones and cones of rings. Using the finite extension property for cones, we answer the question concerning when a compatible partial order of a ring has a compatible linear extension (equivalently, when the positive cone of this order is contained in a full cone). It turns out that, if there is no such extension, then it is caused by a finite system of polynomial-like equations satisfied by some elements of a certain finite subset of the ring and some positive elements.

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 Cohens 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), n ≥ 1, 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 ⟨Z n (R) ∪ C⟩ of R generated by the subset Z n (R) ∪ C of R is also Lie nilpotent of index n.

Which self-maps appear as lattice anti-endomorphisms?

Let f : A → A be a self-map of the set A. We give a necessary and sufficient condition for the existence of a lattice structure (A, ∨, ∧) on A such that f becomes a lattice anti-endomorphism with respect to this structure.

Identities of symmetric and skew-symmetric matrices in characteristicp

AbstractIt 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

A Cayley–Hamilton trace identity for 2×2 matrices over Lie-solvable rings☆☆☆☆☆☆

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

A new class of matrix algebras

is an identity onMn−(Ω), the set ofnxn skew-symmetric matrices over a commutative ring Ω withp1Ω=0 (provided that