Principal submodules in the Schwartz module

In this paper we consider the Schwartz module of entire functions of exponential type having polynomial growth along the real axis. This module is equipped with the non-metrisable locally convex topology. We establish that any principal submodule is the set of limits of the countable converging sequences which members are polynomials multiplied by the generator of the submodule. We also obtain one weak localizability weight criterion for principal submodules and some results concerning with the notion of «synthesizable sequence», which has been recently introduced by A. Baranov and Yu. Belov.