AbstractsMathematics

In this thesis, we study rational extensions of monoids with a zero element. This is essentially a generalisation of the corresponding theory for rings as developed in (7) and (6). (See bibliography at the end). For various reasons, we have decided to consider monoids with a zero element, henceforth called monoids, but most of the results can easily be generalised to general monoids. In the first chapter, we first introduce the notion of an M-set, which is essentially a representation of a monoid, and these M-sets play the same role as R-modules for rings. Then, through the rest of the chapter, we introduce all the tools needed to show that the injective hull of an M-set always exists, and we end the chapter by an example: more specifically, we show that the injective hull of the rationals, considered as a partly ordered set, is the set of reals.