An interesting thing happens when one begins with the axioms of a field, but does not require the associative and commutative properties. The resulting nonassociative division ring is referred to as a ``semifield" in this paper. Semifields have intimate ties to finite projective planes. In short, a finite projective plane with certain restrictions gives rise to a semifield, and, in turn, a finite semifield can be used via a coordinate construction, to build a special finite projective plane. It is also shown that two finite semifields provide a coordinate system for isomorphic projective planes if and only if the semifields are isotopic, where isotopy is a relationship similar to but weaker than isomorphism. Before we prove those results, we explore the nature of isotopy to get a little better feel for the concept. For example, we look at isotopy for associative algebras. We will also examine a particular family of semifields and gather concrete information about solutions to linear equations and isomorphisms. / Master of Science

We investigate commutative semirings, which are formed by a ground set equipped with two binary associative and commutative operations such that one distributes over the other. We narrow down our interest to ideal-simple semirings, that is, semirings without proper ideals. We present the classification of ideal-simple semirings and deal with some classes of ideal-simple semirings, namely semifields and parasemifields. The main result of this thesis is giving tight bounds on the minimal number of generators needed to generate a parasemifield as a semiring. We also study how the semifields that are finitely generated as a semiring look like. Last, but not least, we show that every finitely generated ideal-simple semiring is finitely-generated as a multiplicative semigroup.