In my research I ended up trying to prove some properties of integral domains such that their spectrum is a totally ordered poset. Are there some nice (ubiqitous/natural) examples of such domains, which are not valuation domains? (Of course any local 1-dimensional domain is such, so non-noetherian almost perfect domains form a class of nice examples, so of higher Krull dimension?)



Integral domains with totally ordered spectra
A good source of such rings consist of the so-called "pseudo-valuation domains" of Hedstrom and Houston. A link to their original (1978) article is: <http://projecteuclid.org/download/pdf_1/euclid.pjm/1102810151>. Among other things in that article, they show the following facts that are relevant to your situation:


1. Given any pseudo-valuation domain $R$, there is a unique valuation domain $V$ with $R \subseteq V \subseteq$ the fraction field of $R$, such that the map of prime spectra is an order isomorphism.
2. If you happen to start with a valuation domain of the form $V=K+M$ ($K$ a field, $M$ the maximal ideal of $V$) -- a form that many valuation rings found in nature take -- then take any proper subfield $F$ of $K$ (assuming of course that $K$ isn't a prime field). The subring $R=F+M$ will then be a pseudo-valuation ring whose spectrum is order-isomorphic to that of $V$.


