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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 theideal class group $\Cl(R)$ of a given commutative ring $R$. Among the mainresults, we obtain a general theorem which asserts that for any commutativering $R$ 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, ...

On Grothendieck groups and rings with exact sequences for the Picard, $K_{0}(R)^{\ast}$ and ideal class groups