Jenő Szigeti

New determinants and the Cayley-Hamilton Theorem for matrices over Lie nilpotent rings

We construct the so-called right adjoint sequence of an n × n matrix over an arbitrary ring. For an integer m ≥ 1 the right m-adjoint and the right m-determinant of a matrix is defined by the use of this sequence. Over m-Lie nilpotent rings a considerable part of the classical determinant theory, including the Cayley-Hamilton theorem, can be reformulated for our right adjoints and determinants. The new theory is then applied to derive the PI of algebraicity for matrices over the Grassmann algebra.

Maximal compatible extensions of partial orders

We give a complete description of maximal compatible partial orders on the mono-unary algebra

Linear Orders on General Algebras

(A,f)

Matrix representations of finitely generated Grassmann algebras and some consequences

, where

Determinants for n × n matrices and the symmetric Newton formula in the 3 × 3 case

f:Ato A

Invertibility and Dedekind finiteness in structural matrix rings

is an arbitrary unary operation.

On ideals of triangular matrix rings

We answer the question, when a partial order in a partially ordered algebraic structure has a compatible linear extension. The finite extension property enables us to show, that if there is no such extension, then it is caused by a certain finite subset in the direct square of the base set. As a consequence, we prove that a partial order can be linearly extended if and only if it can be linearly extended on every finitely generated subalgebra. Using a special equivalence relation on the above direct square, we obtain a further property of linearly extendible partial orders. Imposing conditions on the lattice of compatible quasi orders, the number of linear orders can be determined. Our general approach yields new results even in the case of semi-groups and groups.

On the intersection of monotonicity preserving linear extensions

We prove that the m-generated Grassmann algebra can be embedded into a 2m−1×2m−1 matrix algebra over a factor of a commutative polynomial algebra in m indeterminates. Cayley-Hamilton and standard identities for n × n matrices over the m-generated Grassmann algebra are derived from this embedding. Other related embedding results are also presented.

Algebras Generated by Two Quadratic Elements

One of the aims of this paper is to provide a short survey on the -graded, the symmetric and the left (right) generalizations of the classical determinant theory for square matrices with entries in an arbitrary (possibly non-commutative) ring. This will put us in a position to give a motivation for our main results. We use the preadjoint matrix to exhibit a general trace expression for the symmetric determinant. The symmetric version of the classical Newton trace formula is also presented in the case.

Functions realizing as abelian group automorphisms

An example in Szigeti and van Wyk [J. Szigeti and L. van Wyk, Subrings which are closed with respect to taking the inverse, J. Algebra 318 (2007), pp. 1068–1076] suggests that Dedekind finiteness may play a crucial role in a characterization of the structural subrings M n (θ, R) of the full n × n matrix ring M n (R) over a ring R, which are closed with respect to taking inverses. It turns out that M n (θ, R) is closed with respect to taking inverses in M n (R) if all the equivalence classes with respect to θ ∩ θ−1, except possibly one, are of a size less than or equal to p (say) and M p (R) is Dedekind finite. Another purpose of this article is to show that M n (θ, R) is Dedekind finite if and only if M m (R) is Dedekind finite, where m is the maximum size of the equivalence classes (with respect to θ ∩ θ−1). This provides a positive result for the inheritance of Dedekind finiteness by a matrix ring (albeit not a full matrix ring) from a smaller (full) matrix ring.