  In their recent work, Gentili and Struppa proposed a different quaternionic
analogue of the notion of holomorphic functions in the complex plane, called
\textit{slice regular functions}, which has led to several analogues of
classical theorems in complex function theory in the quaternionic setting. The
quaternionic analogue of meromorphic functions is called \textit{semiregular
functions}. Such a function is said to be \textit{slice preserving} if it maps
each domain in a complex line to a domain in the same complex line. In this
work, we deal with the relation between analytic and algebraic properties of
certain fields of semiregular functions in symmetric slice domains of the real
quaternions. More precisely, we prove quaternionic analogues of Bers's theorem
and Iss'sa's theorem that deal with the algebraic structures of the ring of
slice preserving regular functions and of certain fields of slice preserving
semiregular functions on symmetric slice domains in the real quaternions.
