Geodesic
Radicals of semiperfect rings related to idempotents
Fundamentalʹnaâ i prikladnaâ matematika, Tome 6 (2000) no. 1, pp. 293-298

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For a semiperfect ring $A$ we prove the existence of the minimal ideal $\mathcal M(A)$ (modular radical) such that the quotient ring $A/\mathcal M(A)$ has the identity element, and of the minimal ideal $\mathcal W(A)$ (Wedderburn radical) such that the quotient ring $A/\mathcal W(A)$ is decomposable into a direct sum of matrix rings over local rings. A simple criterion of such decomposability is given for left Noetherian semiperfect rings and left perfect rings. 
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     author = {V. T. Markov and A. A. Nechaev},
     title = {Radicals of semiperfect rings related to idempotents},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {293--298},
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     volume = {6},
     number = {1},
     year = {2000},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2000_6_1_a24/}
}
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V. T. Markov; A. A. Nechaev. Radicals of semiperfect rings related to idempotents. Fundamentalʹnaâ i prikladnaâ matematika, Tome 6 (2000) no. 1, pp. 293-298. http://geodesic.mathdoc.fr/item/FPM_2000_6_1_a24/