Geodesic
Primitive skew Laurent polynomial rings
Glasgow mathematical journal, Tome 19 (1978) no. 1, pp. 79-85

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

In [8] the author studied the question of the primitivity of an Ore extension R[x, δ], where δ is a derivation of the ring R. If a is an automorphism of R then it can be shown that R[x, α] is primitive if the following conditions are satisfied: (i) no power αsS ≥ 1, of α is inner; (ii) the only ideals of R invariant under α are 0 and R. These conditions are also known to be necessary and sufficient for the skew Laurent polynomial ring R[x, x−1, α] to be simple [9]. The object of this paper is to find conditions which are sufficient for R[x, x−1, α] to be primitive. The results obtained are remarkably similar to those of [8]. Two logically independent conditions are each found to be sufficient for the primitivity of R[x, x−1, α]. Of these, one is also shown to be sufficient for R[x, α] to be primitive. Included in the examples illustrating these results are some applications to the theory of primitive group rings. The basic techniques involved are also applied to produce a counterexample to the converse of a theorem of Goldie and Michler [3] on when R[x, x−1, α] is a Jacobson ring. 
Jordan, D. A. Primitive skew Laurent polynomial rings. Glasgow mathematical journal, Tome 19 (1978) no. 1, pp. 79-85. doi: 10.1017/S0017089500003414
@article{10_1017_S0017089500003414,
     author = {Jordan, D. A.},
     title = {Primitive skew {Laurent} polynomial rings},
     journal = {Glasgow mathematical journal},
     pages = {79--85},
     year = {1978},
     volume = {19},
     number = {1},
     doi = {10.1017/S0017089500003414},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089500003414/}
}
TY  - JOUR
AU  - Jordan, D. A.
TI  - Primitive skew Laurent polynomial rings
JO  - Glasgow mathematical journal
PY  - 1978
SP  - 79
EP  - 85
VL  - 19
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S0017089500003414/
DO  - 10.1017/S0017089500003414
ID  - 10_1017_S0017089500003414
ER  - 
%0 Journal Article
%A Jordan, D. A.
%T Primitive skew Laurent polynomial rings
%J Glasgow mathematical journal
%D 1978
%P 79-85
%V 19
%N 1
%U http://geodesic.mathdoc.fr/articles/10.1017/S0017089500003414/
%R 10.1017/S0017089500003414
%F 10_1017_S0017089500003414

[1] 1.Bourbaki, N., Commutative algebra (Hermann/Addison-Wesley, 1972). Google Scholar

[2] 2.Cozzens, J. and Faith, C., Simple noetherian rings (Cambridge Univ. Press, 1975). Google Scholar | DOI

[3] 3.Goldie, A. W. and Michler, G., Ore extensions and polycyclic group rings, J. London Math. Soc. (2), 9 (1974), 337–345. Google Scholar | DOI

[4] 4.Hardy, G. H. and Wright, E. M., An introduction to the theory of numbers (4th edition) (Oxford Univ. Press, 1960). Google Scholar

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

[6] 6.Jordan, C. R. and Jordan, D. A., A note on the semiprimitivity of Ore extensions, Comm. Algebra 4 (1976), 647–656. Google Scholar | DOI

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

[8] 8.Jordan, D. A., Primitive Ore extensions, Glasgow Math. J. 18 (1977), 93–97. Google Scholar | DOI

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

[10] 10.Passman, D. S., Primitive group rings, Pacific J. Math. 47 (1973), 499–506. Google Scholar | DOI

Cité par Sources :