  Let $R$ be a Noetherian local ring and $I$ an $R$-ideal. It is well-known
that if the associated graded ring $\gr_I(R)$ is Cohen-Macaulay (Gorenstein),
then so is $R$, but the converse is not true in general. In this paper we
investigate the Cohen-Macaulayness and Gorensteinness of the associated graded
ring $\gr_I(R)$ under the hypothesis of the extended Rees algebra
$R[It,t^{-1}]$ is quasi-Gorenstein or the associated graded ring $\gr_I(R)$ is
a domain.
