No abstract

Let X be a completely regular Hausdor topological space. It is shown that the ring of real valued continuous functions CX is po-coherent (in the sense of F. Wehrung) if and only if X is basically disconnected.(with unknowns x 1 ; . . . ; x n P R) is a ®nitely generated R -subsemimodule of M n;1 r, where M n;1 r denotes the set of all n  1 matrices with entries in R. 1949

The classification of three-dimensional zeropotent algebras over an arbitrary field is given. It is complete up to the individual properties of the ground field.

Let A be an Archimedean uniformly complete f -algebra with unit element. It is shown that A is a po-coherent ring (in the sense of F. Wehrung) if and only if A is Dedekind σ-complete.

Let A be an invertible × complex matrix. It is shown that there is a × permutation matrix P such that the product PA has at least two distinct eigenvalues. The nilpotent complex n × n matrices A for which the products PA with all symmetric matrices P have a single spectrum are determined. It is shown that for a n × n complex matrix A = [a ij ] n i,j= with | i,j a ij | ≤ n n | det A| there exists a permutation matrix P such that the product PA has at least two distinct eigenvalues.