Geodesic
Primitive Ore extensions
Glasgow mathematical journal, Tome 18 (1977) no. 1, pp. 93-97

Voir la notice de l'article provenant de la source Cambridge University Press

Apart from simple Ore extensions such as the Weyl algebras, the best known example of a primitive Ore extension is the universal enveloping algebra U(g) of the 2-dimensional solvable Lie algebra g over a field k of characteristic zero, see [4, p. 22]. U(g) is a polynomial algebra over k in two indeterminates x and y with multiplication subject to the relation xy – yx = y, and may be regarded either as an Ore extension of k [x] by the k-automorphism which maps x to x – 1 or as an Ore extension of k[y] by the derivation yd/dy. The argument suggested in [4, p. 22] to prove the primitivity of U(g) can easily be generalised [6] to show that, if α is an automorphism of the ring R then the following conditions are sufficient for R[x, α] to be primitive: (i) no power αs, s ≧ 1, of α is inner; (ii) the only ideals of R invariant under α are 0 and R. These conditions are necessary and sufficient for the simplicity of the skew Laurent polynomial ring R[x, x–1, α] but are not necessary for the primitivity of R[x, α] (the ordinary polynomial ring D[x] over a division ring D not algebraic over its centre is easily seen to be primitive). 
Jordan, D. A. Primitive Ore extensions. Glasgow mathematical journal, Tome 18 (1977) no. 1, pp. 93-97. doi: 10.1017/S0017089500003086
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[1] 1.Borho, W., Gabriel, P. and Rentschler, R., Primideale in EinhuUenden auflösbarer Lie Algebren, Lecture Notes in Mathematics No. 357 (Springer-Verlag, 1973). Google Scholar | DOI

[2] 2.Eisenbud, D. and Robson, J. C., Modules over Dedekind prime rings, J. Algebra 16 (1970), 67–85. Google Scholar | DOI

[3] 3.Faith, C., Algebra: Rings, modules and categories I (Springer-Verlag, 1973). Google Scholar | DOI

[4] 4.Jacobson, N., Structure of rings (Amer. Math. Soc. Colloquium Publications, rev. edition, 1964). Google Scholar

[5] 5.Jordan, D. A., Noetherian Ore extensions and Jacobson rings, J. London Math. Soc. (2), 10 (1975), 281–291. Google Scholar | DOI

[6] 6.Jordan, D. A., Ph.D. thesis, University of Leeds (1975). Google Scholar

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