Algebraic analysis in structures with the Kaplansky–Jacobson property

D. Przeworska-Rolewicz¹

¹ Institute of Mathematics Polish Academy of Sciences P.O. Box 21, Śniadeckich 8 00-956 Warszawa 10, Poland

Studia Mathematica, Tome 168 (2005) no. 2, pp. 165-186

Résumé

In 1950 N. Jacobson proved that if $u$ is an element of a ring with unit such that $u$ has more than one right inverse, then it has infinitely many right inverses. He also mentioned that I. Kaplansky proved this in another way. Recently, K. P. Shum and Y. Q. Gao gave a new (non-constructive) proof of the Kaplansky–Jacobson theorem for monoids admitting a ring structure. We generalize that theorem to monoids without any ring structure and we show the consequences of the generalized Kaplansky–Jacobson theorem for the theory of linear operators, and even for the classical Calculus. In order to do that, we recall some multiplicative systems, called pseudocategories, very useful in the algebraic theory of perturbations of linear operators. In the second part of the paper, basing on the Kaplansky–Jacobson theorem, we show how to use the above mentioned structures for building Algebraic Analysis of linear operators over a class of linear spaces. We also define (non-linear) logarithmic and antilogarithmic mappings on these structures.

Détail

DOI : 10.4064/sm168-2-7

Keywords: jacobson proved element ring unit has right inverse has infinitely many right inverses mentioned kaplansky proved another recently shum gao gave non constructive proof kaplansky jacobson theorem monoids admitting ring structure generalize theorem monoids without ring structure consequences generalized kaplansky jacobson theorem theory linear operators even classical calculus order recall multiplicative systems called pseudocategories useful algebraic theory perturbations linear operators second part paper basing kaplansky jacobson theorem above mentioned structures building algebraic analysis linear operators class linear spaces define non linear logarithmic antilogarithmic mappings these structures