Geodesic
Rings over which all modules are $I_0$-modules
Fundamentalʹnaâ i prikladnaâ matematika, Tome 13 (2007) no. 5, pp. 193-200.

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Let $A$ be a ring that does not contain an infinite set of idempotents that are orthogonal modulo the ideal $\operatorname{SI}(A_A)$. It is proved that all $A$-modules are $I_0$-modules if and only if either $A$ is a right semi-Artinian right V-ring or $A/\operatorname{SI}(A_A)$ is an Artinian serial ring and the square of the Jacobson radical of $A/\operatorname{SI}(A_A)$ is equal to zero. 
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A. A. Tuganbaev. Rings over which all modules are $I_0$-modules. Fundamentalʹnaâ i prikladnaâ matematika, Tome 13 (2007) no. 5, pp. 193-200. http://geodesic.mathdoc.fr/item/FPM_2007_13_5_a7/

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