| In this thesis we study the embeddability of noncommutative domains into skew fields. First we give a generalization of a famous construction of Malcev (Section 1.3) which shows that not all domains are embeddable into skew fields and point out some related results. A class of semigroups S is constructed so that S is not embeddable into a group, but the semigroup ring D[S] for a domain D is again a domain. We then give a complete proof of Ore's localization theorem for noncommutative domains which we subsequently apply in several locations (Theorem 2.1.3). We also give the construction of Neumann power series rings which constitutes a well-known class of embeddings of group rings over ordered groups into skew fields. We then generalize two constructions of Fisher particularly interested in the height of an embedding of a domain R in a skew field D (see Theorems 4.3.2 and 4.4.8). In particular, embeddings of height one and height two are constructed for free algebras. Finally, in Section 5, we give a proof of Lichtman's theorem and apply it to universal enveloping algebras of Lie algebras to obtain another class of examples of domains embeddable into skew fields. The proof given here appears to be more transparent than the original proof, since sets of representatives are used throughout. |
