Arithmetical rings and quasi-projective ideals

It is proved that a commutative ring $A$ is arithmetical if and only if every finitely generated ideal $M$ of the ring $A$ is a quasi-projective $A$-module and every endomorphism of this module can be extended to an endomorphism of the module $A_A$. These results are proved with the use of some general results on invariant arithmetical rings.