Geodesic
A remark on commutative arithmetic rings
Fundamentalʹnaâ i prikladnaâ matematika, Tome 19 (2014) no. 2, pp. 21-23 
E. S. Golod. A remark on commutative arithmetic rings. Fundamentalʹnaâ i prikladnaâ matematika, Tome 19 (2014) no. 2, pp. 21-23. http://geodesic.mathdoc.fr/item/FPM_2014_19_2_a1/
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Voir la notice de l'article provenant de la source Math-Net.Ru

It is proved that a commutative ring with identity $R$ is arithmetic (i.e., the ideal lattice of $R$ is distributive) if and only if for any finitely generated (or any finitely presented) $R$-module $M$ and any ideal $I$ of $R$ the equality $I+\operatorname{Ann}M=\operatorname{Ann}(M/IM)$ holds.

[1] Tuganbaev A. A., Teoriya kolets. Arifmeticheskie moduli i koltsa, MTsNMO, M., 2009