In terms of the duality property of injective preenvelopes and flat
precovers, we get an equivalent characterization of left Noetherian rings. For
a left and right Noetherian ring $R$, we prove that the flat dimension of the
injective envelope of any (Gorenstein) flat left $R$-module is at most the flat
dimension of the injective envelope of $_RR$. Then we get that the injective
envelope of $_RR$ is (Gorenstein) flat if and only if the injective envelope of
every Gorenstein flat left $R$-module is (Gorenstein) flat, if and only if the
injective envelope of every flat left $R$-module is (Gorenstein) flat, if and
only if the (Gorenstein) flat cover of every injective left $R$-module is
injective, and if and only if the opposite version of one of these conditions
is satisfied.