Geodesic
Criteria of semisimplicity of skew polynomial ring
Fundamentalʹnaâ i prikladnaâ matematika, Tome 1 (1995) no. 3, pp. 701-709

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Let $R$ be an associative ring and $f$ be an injective endomorphism of $R$ such that the Cohn–Jordan extension $A(R,f)$ satisfies the ascending chain condition on left annihilators. In this paper we obtain some semiprimitivity criteria for the skew polynomial ring $R[x,f]$ over the ring $R$. In particular, we prove that the skew polynomial ring is semisimple if and only if its prime radical is zero. Furthermore, it is so if and only if the ring $R$ is semiprime. 
@article{FPM_1995_1_3_a8,
     author = {V. A. Mushrub},
     title = {Criteria of semisimplicity of skew polynomial ring},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {701--709},
     publisher = {mathdoc},
     volume = {1},
     number = {3},
     year = {1995},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_1995_1_3_a8/}
}
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V. A. Mushrub. Criteria of semisimplicity of skew polynomial ring. Fundamentalʹnaâ i prikladnaâ matematika, Tome 1 (1995) no. 3, pp. 701-709. http://geodesic.mathdoc.fr/item/FPM_1995_1_3_a8/