Geodesic
Rings All of whose Factor Rings are Semi-Prime
Canadian mathematical bulletin, Tome 12 (1969) no. 4, pp. 417-426

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We prove in this paper that fifteen classes of rings coincide with the class of rings named in the title. One of them is the class of rings R such that X2 = X for each R-ideal X: we shall refer to rings with this property (and thus to the rings of the title) as fully idempotent rings. The simple rings and the (von Neumann) regular rings are fully idempotent. Indeed, every finitely generated right or left ideal of a regular ring is generated by an idempotent [l, p. 42], so that X2 = X holds for every one-sided ideal X. 
Courter, R.C. Rings All of whose Factor Rings are Semi-Prime. Canadian mathematical bulletin, Tome 12 (1969) no. 4, pp. 417-426. doi: 10.4153/CMB-1969-052-2
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     title = {Rings {All} of whose {Factor} {Rings} are {Semi-Prime}},
     journal = {Canadian mathematical bulletin},
     pages = {417--426},
     year = {1969},
     volume = {12},
     number = {4},
     doi = {10.4153/CMB-1969-052-2},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1969-052-2/}
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