  Let $R$ be a commutative ring. An $R$-module $M$ is said to be super finitely
presented if there is an exact sequence of $R$-modules $\cdots\rightarrow
P_n\rightarrow\cdots \rightarrow P_1\rightarrow P_0\rightarrow M\rightarrow 0$
where each $P_i$ is finitely generated projective. In this paper it is shown
that if $R$ has the property (B) that every super finitely presented module has
finite Gorenstein projective dimension, then every finitely generated
Gorenstein projective module is super finitely presented. As an application of
the notion of super finitely presented modules, we show that if $R$ has the
property (C) that every super finitely presented module has finite projective
dimension, then $R$ is $K_0$-regular, i.e., $K_0(R[x_1,\cdots,x_n])\cong
K_0(R)$ for all $n\geq 1$.
