This introductory talk will address an aspect of the problem of embedding rings into skew fields (that is, division rings). For commutative rings we have a straightforward resolution: a commutative ring embeds into a field if and only if it is a domain (does not have zero divisors); moreover, in the latter case it has a unique field of fractions. However, the situation is grimmer for noncommutative rings. There exist noncommutative domains without embeddings into any skew fields and in general there is no "simple" necessary and sufficient condition for embeddability as in the commutative setting. If a ring embeds into a skew field, then we can talk about its skew fields of fractions. Namely, a skew field U is a skew field of fractions of a ring R if R is its subring and R generates U as a skew field. Contrary to the commutative case, a noncommutative ring R may admit various non-isomorphic skew fields of fractions. For this reason we are usually interested in a distinguished one, which we call the universal skew field of fractions of R. For example, free algebras and their tensor products admit universal skew field of fractions and we will briefly describe their constructions. |