In the present paper, we will show that the set of minimal elements of a full affine semigroup $A\hookrightarrow {\mathbb{N}}_0^k$ contains a free basis of the group generated by $A$ in ${\mathbb{Z}}^k$. This will be applied to the study of the group ${\text{K}}_0\left(R\right)$ for a semilocal ring $R$.

We first prove that every countably presented module is a pure epimorphic image of a countably generated pure-projective module, and by using this we prove that if every countably generated pure-projective module is pure-injective then every module is pure-injective, while if in any countably generated pure-projective module every countably generated pure-projective pure submodule is a direct summand then every module is pure-projective.