L. van Wyk

Do isomorphic structural matrix rings have isomorphic graphs

We first provide an example of a ring R such that all possible 2 × 2 structural matrix rings over R are isomorphic. However, we prove that the underlying graphs of any two isomorphic structural matrix rings over a semiprime Noetherian ring are isomorphic, i.e. the underlying Boolean matrix B of a structural matrix ring M(B,R) over a semiprime Noetherian ring R can be recovered, contrary to the fact that in general R cannot be recovered.

Isomorphism of generalized triangular matrix-rings and recovery of tiles

We prove an isomorphism theorem for generalized triangular matrix-rings, over rings having only the idempotents 0and 1, in particular, over indecomposable commutative rings or over local rings (not necessarily commutative). As a consequence, we obtain a recovery result for the tile in a tiled matrix-ring.

The J2-radical in structural matrix near-rings

Abstract We show that the two obvious definitions for a structural matrix near-ring somewhat unexpectedly yield the same near-ring. The strictly maximal left ideals of a structural matrix near-ring are characterized and used to describe its J 2 -radical.

Isomorphisms between Morita context rings

Let (R, S, R M S , S N R , f, g) be a general Morita context, and let be the ring associated with this context. Similarly, let be another Morita context ring. We study the set Iso(T, T ′) of ring isomorphisms from T to T ′. Our interest in this problem is motivated by: (i) the problem to determine the automorphism group of the ring T, and (ii) the recovery of the non-diagonal tiles problem for this type of generalized matrix rings. We introduce two classes of isomorphisms from T to T ′, the disjoint union of which is denoted by Iso0(T, T ′). We describe Iso0(T, T ′) by using the ℤ-graded ring structure of T and T ′. Our main result characterizes Iso0(T, T ′) as the set consisting of all semigraded isomorphisms and all anti-semigraded isomorphisms from T to T ′, provided that the rings R′ and S′ are indecomposable and at least one of M′ and N′ is nonzero; in particular, Iso0(T, T ′) contains all graded isomorphisms and all anti-graded isomorphisms from T to T ′. We also present a situation where Iso0(T, T ′) = Iso(T, T ′). This is in the case where R, S, R′ and S′ are rings having only trivial idempotents and all the Morita maps are zero. In particular, this shows that the group of automorphisms of T is completely determined.

On the small and essential ideals in certain classes of rings

It is well known that for a ring with identity the Brown-McCoy radical is the maximal small ideal. However, in certain subrings of complete matrix rings, which we call structural matrix rings, the maximal small and minimal essential ideals coincide. In this paper we characterize a class of commutative and a class of non-commutative rings for which this coincidence occurs, namely quotients of Prufer domains and structural matrix rings over Brown-McCoy semisimple rings. A similarity between these two classes is obtained.

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

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.

Semigroup gradings of full matrix rings.

Let F be a field, and let A = Mn (F) be the algebra of all n × n matrices over F. For each finite semigroup S, we describe all gradings A = ⊕ s∈S A S of A by S such that all standard matrix units eij of A are homogeneous elements of A.

Complete blocked triangular matrix rings over a Noetherian ring

Abstract We show that the underlying Boolean matrix B of a complete blocked triangular matrix ring M (B, R) over a left Noetherian ring R is unique, i.e. if M (B 1 , R) and M (B 2 , R) are isomorphic complete blocked triangular matrix rings over a left Noetherian ring R , then B 1 = B 2 .

Surface conductivity as an indication of the surface condition of non-ceramic insulators

Different outdoor insulating materials were exposed to marine pollution conditions and their surface conductivities were measured using an insulator pollution monitoring apparatus (IPMA), originally designed to measure surface conductivity on ceramic insulators. The current and conductivity values were used as a criterion to evaluate the different materials. EPDM with different creepages had dissimilar performances while silicone rubber insulators compared to EPDM and glass proved to be superior in having a lower surface conductivity.

Near-Rings of Invariants

abstractFor a group S acting on a group G we let I(S,G) = {ƒ: G → G ƒ σ(x) = ƒ (x), ∈ G, σ ∈ S}, the near-ring of invariants of (S,G). In this paper we investigate the transfer of information between the ideal structure of I(S,G) and the group action (S, G).