Geodesic
Rings without infinite sets of noncentral orthogonal idempotents
Fundamentalʹnaâ i prikladnaâ matematika, Tome 14 (2008) no. 2, pp. 207-221

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Let $A$ be a ring without infinite sets of noncentral orthogonal idempotents. $A$ is an exchange ring if and only if all Pierce stalks of $A$ are semiperfect rings. All $A$-modules are $I_0$-modules if and only if either $A$ is a right semi-Artinian ring in which every proper right ideal is the intersection of maximal right ideals or $A/\operatorname{SI}(A_A)$ is an Artinian serial ring such that the square of the Jacobson radical of $A/\operatorname{SI}(A_A)$ is equal to zero. 
@article{FPM_2008_14_2_a9,
     author = {A. A. Tuganbaev},
     title = {Rings without infinite sets of noncentral orthogonal idempotents},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {207--221},
     publisher = {mathdoc},
     volume = {14},
     number = {2},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2008_14_2_a9/}
}
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A. A. Tuganbaev. Rings without infinite sets of noncentral orthogonal idempotents. Fundamentalʹnaâ i prikladnaâ matematika, Tome 14 (2008) no. 2, pp. 207-221. http://geodesic.mathdoc.fr/item/FPM_2008_14_2_a9/