The generalized spectral decomposition expressing a linear operator as a sum of idempotent and nilpotent operators is derived from the partial fraction decomposition of the reciprocal of its minimal polynomial, and makes possible a functional calculus of analytic functions of the operator. The classical spectral and polar decompositions for matrices are shown to be easy consequences. The idempo...

Abstract We associate a simplicial complex to a sequence of ring elements that generate the ring over R (or nearly so). This is motivated by a formula for the degree of R as the “volume” of the simplicial complex. Then we give a flat degeneration of the ring to one combinatorially defined from the simplicial complex – or almost; the simplicial complex sometimes has to be promoted to a simplicia...

Let R be a ring with Jacobson ideal J and center C. McCoy and Montgomery introduced the concept of a p-ring (p prime) as a ring R of characteristic p such that xp = x for all x in R. Thus, Boolean rings are simply 2-rings (p = 2). It readily follows that a p-ring (p prime) is simply a ring R of prime characteristic p such that R ⊆ N + Ep, where N = {0} and Ep = {x ∈ R : xp = x}. With this as mo...

We consider in this article a Bernstein property of probability measures on groups introduced by Neuenschwander. We discuss this property for discrete groups, compact groups, nilpotent groups and some solvable groups. In all these cases we show that a measure having the Bernstein-Neuenschwander property must be concentrated on an Abelian subgroup. We conclude with an application of this result ...