Living polymers formed by reversible association of spacers (oligomers) terminated by one donor and one acceptor group at the ends are studied by means of Monte Carlo simulations (using the bond-fluctuation model). To account for the different chemical nature of the associating groups and spacers, we considered three cases of flexible, semiflexible, and rigid polymers. Rigid polymers have both intrinsic rigidity of spacers and rigidity (entropic penalty) imparted by end groups association. Semiflexible polymers possess only the latter, and flexible polymers do not have any type of rigidity. We have studied the average degree of association for all types of polymers as a function of concentration and spacer length and compared the results with a simple analytical model. We found that above some crossover concentration the association behaviors of all chains follow the same pattern, well-described by the analytical model and depending mainly on the number density of donor/acceptor groups. Below the crossover concentration, the association is governed by the individual characteristics of spacers and differs for chains of different rigidity and chain length. Defined in this way the crossover concentration, c(cr), is a measure of the ring-to-chain transition: below c(cr) ring formation is most favorable, while above c(cr) linear chain formation dominates. We found that the rigidity (entropic penalty) imparted by end group association defines the behavior above the c(cr): the total degree of association of semiflexible chains is much smaller than that for flexible chains of the same concentration and practically coincides with that for rigid chains (despite different spacer rigidity). The crossover concentration, c, is a function of spacer length and strongly differs for chains of different rigidity: c(cr)(rigid) much less than c(cr)(semi) < c(cr)(flex). The rigidity of the spacer has a much stronger impact on c(cr) compared to the end group rigidity imparted by association.