  The main goal of this article is to investigate the Grothendieck groups,
especially the Grothendieck ring $K_{0}(R)$, the Picard group $\Pic(R)$ and the
ideal class group $\Cl(R)$ of a given commutative ring $R$. Among the main
results, we obtain a general theorem which asserts that for any commutative
ring $R$ we have the following exact sequence of groups:
$$\xymatrix{0\ar[r]&\Cl(R)\ar[r]&\Pic(R)\ar[r]&\Pic(T(R))}$$ where $T(R)$
denotes the total ring of fractions of $R$. As an application, ...
