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Primitive Ore extensions

Published online by Cambridge University Press:  18 May 2009

D. A. Jordan
D. A. Jordan
Affiliation:
Department of Pure Mathematics, University of Sheffield, Sheffield S10 2TN
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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).

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 18 , Issue 1 , January 1977 , pp. 93 - 97
Copyright
Copyright © Glasgow Mathematical Journal Trust 1977

References

REFERENCES

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