Ukrainian Mathematical Bulletin

Volume 1, Issue 1, 2004 pp. 137-148.

Pseudo-Nearrings and Quasi-Modules over Them

Authors: Anna Chwastyk and Kazimierz Glazek
Author institution: Institute of Mathematics, Technical University of Opole, Institute of Mathematics, Opole, Poland, Institute of Mathematics, University of Zielona Ga'ora, Zielona Ga'ora, Poland

Summary: In this paper, we begin the investigation of a new notion of pseudo-nearrings and a generalization of linear spaces to quasi-modules over pseudo-nearrings. Pseudo-nearrings can be treated as ringoids in the sense of J.~Hion (see [6]). The idea of pseudo-nearrings is based on the notion of $^*$-associative quasigroup, i.\,e., on an involutive groupoid $(A;+,^{\ast }) $ in which the following identities hold: \[ (x^{\ast})^{\ast}=x,\ (x+y)^{\ast}=y^{\ast}+x^{\ast},\, (x+y)^{\ast}+z=x+(y+z)^{\ast}. \] We also assume the commutativity and quasigroup properties of $(A;+)$.

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