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. 
@article{FPM_2008_14_2_a0,
     author = {A. N. Abyzov and A. A. Tuganbaev},
     title = {Rings over which all modules are $I_0${-modules.~II}},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {3--12},
     publisher = {mathdoc},
     volume = {14},
     number = {2},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2008_14_2_a0/}
}
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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/