  The Galois ring $GR(p^r,m)$ of characteristic $p^r$ and cardinality $p^{rm}$,
where $p$ is a prime and $r,m \ge 1$ are integers, is a Galois extension of the
residue class ring $Z_{p^r}$ by a root $\omega$ of a monic basic irreducible
polynomial of degree $m$ over $Z_{p^r}$. Every element of $GR(p^r,m)$ can be
expressed uniquely as a polynomial in $\omega$ with coefficients in $Z_{p^r}$
and degree less than or equal to $m-1$, thus $GR(p^r,m)$ is a free module of
rank $m$ over $Z_{p^r}$ with basis $\{1,\omega, \omega^2,..., \omega^{m-1} \}$.
The ring $Z_{p^r}$ satisfies the invariant dimension property, hence any other
basis of $GR(p^r,m)$, if it exists, will have cardinality $m$.
  This paper was motivated by the code-theoretic problem of finding the
homogeneous bound on the $p^r$-image of a linear block code over $GR(p^r,m)$
with respect to any basis. It would be interesting to consider the dual and
normal bases of $GR(p^r,m)$.
  By using a Vandermonde matrix over $GR(p^r,m)$ in terms of the generalized
Frobenius automorphism, a constructive proof that every basis of $GR(p^r,m)$
has a unique dual basis is given. The notion of normal bases was also
generalized from the classic case for Galois fields.
