J.F. Watters

Normal radicals and normal classes of rings

Abstract A radical N in the category of rings is called normal if, for any Morita context (R, V, W, S), we have VN(S)W ⊆ N(R). In this paper these radicals are investigated and the related notion of a normal class of prime rings is defined. A characterization of normal, special radicals is given and it is shown that normal classes generalize special classes in a natural way. Several results are given on the closure of normal classes under forming related rings and some theorems on structure spaces are extended.

Simultaneous quasi-diagonalization of normal matrices

Abstract We show that if S is a family of n×n normal matrices and As is the algebra generated by S over the complex field, then the matrices in S are simultaneously unitarily similar to quasi-diagonal matrices if and only if (AB − BA)2Q = Q(AB − BA)2 for all A and B ∈ As, Q ∈ S. In fact, the domain of B can be further restricted. For the purposes of this paper a quasi-diagonal matrix will be a matrix of block diagonal form dg(D1,D2,…,Dk) with each Di either 1×1 or 2×2, and zeros elsewhere.

Block Triangularization of Algebras of Matrices

Abstract This paper is concerned with the interdependence of the irreducible constituents of an algebra of n × n matrices over a field F . It is shown that there is a similarity transformation reducing the algebra to a block triangular form in which, ateach pair of diagonal places, the blocks either are always equal or may be occupied by any entries from the corresponding irreducible constituents. A recent theorem of Kaplan-sky is extended as an application of this result.

Universally reflexive algebras

Finite dimensional algebras whose representations are all reflexive are characterized as those over which the homomorphisms between any pair of indecomposable projective modules form a vector space of dimension at most one. This includes incidence algebras over finite preordered sets.

Algebras whose projective modules are reflexive

The algebras of the title are investigated. They are shown to include l-hereditary algebras, monomial algebras whose indecomposable projective representations are reflexive, and certain binomial algebras.

Classes of simple modules and triangular rings

Given any isomorphically closed class of simple modules over a ring R analogues of the Jacobson radical and socle are studied. A triangular decomposition theorem is proved (Theorem I), in the case when R is contains no infinite family of orthogonal idempotents, and this includes an analogue of a Theorem of Gordon. We also provide in Lemma 4 a description of this socle in a triangular matrix ring when the class is the class of all simple projective modules. Finally, a structure theorem IS proved for the rings R, with no infinite family of orthogonal idempotents, in which gRg has a projective simple module for all idempotents g with gR(l-g) = 0.

Direct sums of reflexive modules

Abstract The problem of determining when reflexivity is inherited by direct sums of reflexive modules over finite dimensional split algebras is addressed. We show that this holds if the algebra is left locally distributive, a class of algebras which includes serial algebras, as well as for those algebras R with radical J and a basic set of idempotents such that dim( eJf ) ≤ 3 for all idempotents e , f in the set. In the opposite direction, it is proved that a necessary condition for direct sums of reflexive modules to be reflexive is that the quiver of the algebra should contain no triple arrows.

Loewy series, V-modules and trace ideals

If R Vis a V-module and (R V W S) is a Morita context in which (S/WV) s is flat, then the trace ideal WVis left V-module over S. If, in additionS:(S/WV) is flat and S/WVis a fully left idempotent ring, then Sis also fully left idempotent. The lower (upper) Loewy length of R Vprovides an upper bound for the corresponding Loewy length of s(WV).