Pseudo-nearrings and quasi-modules over them

Abstract

In this paper we start to investigate 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-nearings is based on the notion of a ∗-associative quasigroup, i.e. on an involutive groupoid (A;+,* ) in which the following identities hold: (x*)* = x, (x + y)* = y* + x*, (x + y)* + z = x + (y + z)*. We assume also commutativity and quasigroup properties of (A;+).