Leon van Wyk

An internal characterisation of structural matrix rings

It is known that structural matrix rings pro-vide a natural passage from complete matrix rings to upper and lower triangular matrix rings, and they often explain the peculiarities regarding certain properties of complete matrix rings on the one hand and of triangular matrix rings on the other hand. In this paper the concept of a set of matrix units in a ring associated with a quasi-order relation is introduced and used to provide an internal char-acterisation of structural matrix rings.

Matrix rings satisfying column sum conditions versus structural matrix rings

Abstract The internal characterization of a structural matrix ring in terms of a set of matrix units associated with a quasiorder relation is used to obtain isomorphisms between seemingly different classes of subrings of a complete matrix ring.

Matrix representations of finitely generated Grassmann algebras and some consequences

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.

MATRIX RINGS OF INVARIANT SUBSPACES

Abstract Consider a set V of R-submodules of R n, R a commutative ring. We provide a sufficient condition on V such that the ring of all R-endomorphisms of R n, which leave every R-submodule in V invariant, is a structural matrix ring. We also obtain results suggesting strongly that the mentioned condition is necessary too in case R is a field.

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.

Invertibility and Dedekind finiteness in structural matrix rings

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.

On ideals of triangular matrix rings

We provide a formula for the number of ideals of complete block-triangular matrix rings over any ring R such that the lattice of ideals of R is isomorphic to a finite product of finite chains, as well as for the number of ideals of (not necessarily complete) block-triangular matrix rings over any such ring R with three blocks on the diagonal.

When is a centralizer near-ring isomorphic to a matrix near-ring? Part 2

Necessary conditions are found for a centralizer near-ring MA(G) to be isomorphic to a matrix near-ring, where G is a finite group which is cyclic as an MA(G)-module There are centralizer near-rings which are matrix near-rings. A class of such near-rings is exhibited. Examples of centralizer near-rings which are not matrix near-rings are given.

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 ⟨x, y | x 2 + ax + b = 0, y 2 + cy + d = 0⟩ for suitable a, b, c, d ∈ K. We establish that F can be embedded into the 2 × 2 matrix algebra with entries from the polynomial algebra over the algebraic closure of K and that F and 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 1980s. When each of the equations has different zeros, we improve a result of Weiss, also from the 1980s.

Lie solvability in matrix algebras

ABSTRACT If an algebra satisfies the polynomial identity (for short, is ), then is trivially Lie solvable of index (for short, is ). We prove that the converse holds for subalgebras of the upper triangular matrix algebra any commutative ring, and . We also prove that if a ring S is (respectively, ), then the subring of comprising the upper triangular matrices with constant main diagonal, is (respectively, ) for all . We also study two related questions, namely whether, for a field F, an subalgebra of for some n, with (F-)dimension larger than the maximum dimension of a subalgebra of , exists, and whether a subalgebra of with (the mentioned) maximum dimension, other than the typical subalgebras of with maximum dimension, which were described by Domokos and refined by van Wyk and Ziembowski, exists. Partial results with regard to these two questions are obtained.