  Let $K$ be a field and $R=\oplus_{p\in\mathbb{N}}R_p$ an $\mathbb{N}$-graded
$K$-algebra, which has an SM $K$-basis (i.e. a skew multiplicative $K$-basis)
such that $R$ holds a Gr\"obner basis theory. It is proved that there is a
one-to-one correspondence between the set of Gr\"obner bases in $R$ and the set
of dh-closed homogeneous Gr\"obner bases in the polynomial algebra $R[t]$; and
that the similar result holds true if $R$ and $R[t]$ are replaced respectively
by the free algebra $K< X_1,...,X_n>$ and the free algebra $K< X_1,...,X_n,T>$.
Moreover, it is shown that dh-closed graded ideals in $R[t]$ and $K<
X_1,...,X_n, T>$ can be realized by dh-closed homogeneous Gr\"obner bases. The
latter result indeed tells us that algebras defined by dh-homogeneous Gr\"obner
bases can be studied as Rees algebras effectively via more simpler algebras as
demonstrated in ([7], [8]).
