Extended Algebraic Structure of Quasimodules

A Module is one of the common and significant algebraic structures of modern algebra. We have introduced in our paper “An Associated Structure of a Module” published in Revista de la Academia Canaria de Ciencias, Volume XXV, 9–22 (2013), the concept of a quasimodule which is a generalisation of a module that speaks of a topological hyperspace structure as well as a module structure in some sense. A quasimodule is a conglomeration of semigroup structure, ring multiplication and partial order; this structure always contains a module. In the present paper we shall introduce the concept of an ideal in a quasimodule. This concept is completely different from the concept of an ideal in a ring. We shall discuss several properties of ideals and construct the ideal generated by any subset of a quasimodule. We shall define a minimal ideal and find a necessary and sufficient condition for a proper ideal to be a minimal ideal.