Subperiodic Rings with Conditions on Extended Commutators

Remarks on the Commutativity of Rings

Introduction. A celebrated theorem of N. Jacobson [7] asserts that if (1) x*(x) =x for every x in a ring R, where n(x) is an integer greater than one, then R is commutative. In a recent paper [2], I. N. Herstein has shown that it is enough to require that (1) holds for those x in R which are commutators: x= [y, z]=yz — zy of two elements of R. The purpose of this note is to show that if R has n...

A History of Selected Topics in Categorical Algebra I: From Galois Theory to Abstract Commutators and Internal Groupoids

This paper is a chronological survey, with no proofs, of a direction in categorical algebra, which is based on categorical Galois theory and involves generalized central extensions, commutators, and internal groupoids in Barr exact Mal’tsev and more general categories. Galois theory proposes a notion of central extension, and motivates the study of internal groupoids, which is then used as an a...

Vanishing Power Values of Commutators with Derivations on Prime Rings

From subperiodic to low cardinality quadrature formulas for noncircular pupils

We discuss numerical integration formulas with polynomial exactness for noncircular pupils, based on subperiodic trigonometric Gaussian quadrature and compression of discrete measures. Such formulas can be useful in optical design (numerical ray-tracing in obscured and vignetted pupils for accurate spot size computation), and in optical diffraction theory under conditions of partial coherence (...

Nilpotent Elements in Skew Polynomial Rings

 Letbe a ring with an endomorphism and an -derivationAntoine studied the structure of the set of nilpotent elements in Armendariz rings and introduced nil-Armendariz rings. In this paper we introduce and investigate the notion of nil--compatible rings. The class of nil--compatible rings are extended through various ring extensions and many classes of nil--compatible rings are constructed. We al...