  We study masses and decay widths of positive and negative parity nucleons
using a three-flavor parity doublet model, in which we introduce three
representations, $\left[({\bf 3} , \bar{{\bf 3}})\oplus (\bar{{\bf 3}} , {\bf
3})\right]$, $\left[({\bf 3} , {\bf 6}) \oplus ({\bf 6}, {\bf 3})\right]$, and
$\left[({\bf 8} , {\bf 1}) \oplus ({\bf 1} , {\bf 8})\right]$ of the chiral
U$(3)_{\rm L}\times$U$(3)_{\rm R}$ symmetry. We find an extended version of the
Goldberger-Treiman relation among the mass differences and the coupling
constants for pionic transitions. This relation leads to an upper bound for the
decay width of $N(1440) \rightarrow N(939) + \pi$ independently of the model
parameters. We perform the numerical fitting of the model parameters and derive
several predictions, which can be tested in future experiments or lattice QCD
analysis. Furthermore, when we use the axial coupling of the excited nucleons
obtained from lattice QCD analyses as inputs, we find that the ground state
nucleon $N(939)$ consists of about 80% of $\left[({\bf 3} , {\bf 6}) \oplus
({\bf 6}, {\bf 3})\right]$ component, and that the chiral invariant mass of
$N(939)$ is roughly $500$ -- $800$MeV.
