Geodesic
On rings radical over commutative subrings
Sbornik. Mathematics, Tome 12 (1970) no. 4, pp. 511-520 
A. I. Likhtman. On rings radical over commutative subrings. Sbornik. Mathematics, Tome 12 (1970) no. 4, pp. 511-520. http://geodesic.mathdoc.fr/item/SM_1970_12_4_a1/
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     author = {A. I. Likhtman},
     title = {On rings radical over commutative subrings},
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     volume = {12},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1970_12_4_a1/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

The following theorem is proved. If a ring $R$ is radical over a commutative subring $K$, then all the nilpotent elements of $R$ generate a null-ideal $T$ for which the corresponding factor ring is commutative. An affirmative answer is thus provided for a question raised by Faith. Bibliography: 5 titles.

[1] C. Faith, “Algebraic division ring extensions”, Proc. Amer. Math. Soc., 11 (1960), 43–53 | DOI | MR | Zbl

[2] C. Faith, “Radical extensions of ring”, Proc. Amer. Math. Soc., 12 (1961), 274–283 | DOI | MR | Zbl

[3] H. Dzhekobson, Stroenie kolets, IL, Moskva, 1961

[4] E. P. Armendaris, “On radical extensions of rings”, J. Australian Math. Soc., 7:4 (1967), 552–554 | DOI | MR

[5] N. Jacobson, Lectures in abstract algebra, v. 1, New York, 1951