ABSTRACT
This chapter determines some conditions under which a cancellative semigroup has a group of right fractions. It establishes the existence of a division ring of quotients of a domain that has no free noncommutative subalgebras. The chapter extends the class of semigroups for which the group of fractions is known to exist. It lists important cases in which cancellative semigroup has a group of right fractions. The chapter presents advantages coming from the fact that cancellative semigroup has a group of (right) fractions to exploit the group algebra results. It also discusses the observations made on the behavior of primeness and semiprimeness under localizations.
