Geodesic
Rings over which all modules are $I_0$-modules.~II
Fundamentalʹnaâ i prikladnaâ matematika, Tome 14 (2008) no. 2, pp. 3-12.

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All right $R$-modules are $I_0$-modules if and only if either $R$ is a right SV-ring or $R/I^{(2)}(R)$ is an Artinian serial ring such that the square of the Jacobson radical of $R/I^{(2)}(R)$ is equal to zero. 
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A. N. Abyzov; A. A. Tuganbaev. Rings over which all modules are $I_0$-modules.~II. Fundamentalʹnaâ i prikladnaâ matematika, Tome 14 (2008) no. 2, pp. 3-12. http://geodesic.mathdoc.fr/item/FPM_2008_14_2_a0/

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