  Replacing $\{0\}$ by the whole ideal of infinitesimals yields a weaker notion
of \emph{archimedean element} that we call \emph{quasiarchimedean}. It is known
that semisimple MV-algebras with compact maximal spectrum (in the co-Zarisky
topology) are exactly the hyperarchimedean algebras. We characterise all the
algebras with compact maximal spectrum as being \emph{quasihyperarchimedean}
\mbox{MV-algebras,} which in a sense are non semisimple hyperarchimedean
algebras. We develop some basic facts in the theory of MV-algebras along the
lines of algebraic geometry, where infinitesimals play the role of nilpotent
elements, and prove a MV-algebra version of Hilbert's Nullstellensatz. Finally
we consider the relations (some inedited) between several elementary classes of
MV-algebras in terms of the ideals that characterise them, and present
elementary (first order with denumerable disjunctions) proofs in place of the
\mbox{set-theoretical} usually found in the literature.
