Characterizing nearly simple chain domains

G. Puninski, using model theoretical methods, showed that if a chain domain $R$ is nearly simple, then $Ra+bR = J(R)$ for any nonzero elements $a,b$ in $J(R)$, the Jacobson radical of $R$. Here, an algebraic proof is given for this result, exceptional chain domains are characterized, and it is shown that $V_0(R)$, the lattice generated by all proper nonzero left and right ideals, is a direct product of two linearly ordered sets if $R$ is nearly simple. In a certain sense this property characterizes nearly simple chain domains among all integral domains.