Geodesic
Localization and Completion at Primes Generated by Normalizing Sequences in Right Noetherian Rings
Canadian journal of mathematics, Tome 33 (1981) no. 2, pp. 325-346

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

If P is a right localizable prime ideal in a right Noetherian ring R, it is known that the ring RP is right Noetherian, that its Jacobson radical is the only maximal ideal, and that RP /J(RP) is simple Artinian: in short it has several properties of the commutative local rings.In the present work we examine the properties of RP under the additional assumption that P is generated by, or is a minimal prime above, a normalizing sequence. It is shown that in such cases J(RP) satisfies the AR-property (i.e., P is classical) and that the rank of P coincides with the Krull dimension of RP. The length of the normalizing sequence is shown to be an upper bound for the rank of P, and if P is generated by a normalizing sequence x 1, x 2, ..., xn then the rank of P equals n if and only if the P-closures of the ideals Ij generated by x 1, x 2, ..., xj (j = 0, 1, ..., n), are all distinct primes. 
Heinicke, A. G. Localization and Completion at Primes Generated by Normalizing Sequences in Right Noetherian Rings. Canadian journal of mathematics, Tome 33 (1981) no. 2, pp. 325-346. doi: 10.4153/CJM-1981-027-3
@article{10_4153_CJM_1981_027_3,
     author = {Heinicke, A. G.},
     title = {Localization and {Completion} at {Primes} {Generated} by {Normalizing} {Sequences} in {Right} {Noetherian} {Rings}},
     journal = {Canadian journal of mathematics},
     pages = {325--346},
     year = {1981},
     volume = {33},
     number = {2},
     doi = {10.4153/CJM-1981-027-3},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1981-027-3/}
}
TY  - JOUR
AU  - Heinicke, A. G.
TI  - Localization and Completion at Primes Generated by Normalizing Sequences in Right Noetherian Rings
JO  - Canadian journal of mathematics
PY  - 1981
SP  - 325
EP  - 346
VL  - 33
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1981-027-3/
DO  - 10.4153/CJM-1981-027-3
ID  - 10_4153_CJM_1981_027_3
ER  - 
%0 Journal Article
%A Heinicke, A. G.
%T Localization and Completion at Primes Generated by Normalizing Sequences in Right Noetherian Rings
%J Canadian journal of mathematics
%D 1981
%P 325-346
%V 33
%N 2
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1981-027-3/
%R 10.4153/CJM-1981-027-3
%F 10_4153_CJM_1981_027_3

[1] 1. Cohn, P. M., Skew field constructions (Cambridge University Press, Cambridge, London, New York, Melbourne, 1977). Google Scholar

[2] 2. Cozzens, J. H. and Sandomierski, F. L., Localization at a semiprime ideal of a right Noetherian ring, Comm. Algebra 5 (1977), 707–726. Google Scholar

[3] 3. Deshpande, V. K., Completions of Noetherian hereditary prime rings, Pac. J. Math. 90 (1980), 285–325. Google Scholar

[4] 4. Gordon, R. and Robson, J. C., Krull dimension, Mem. Amer. Math. Soc. 133 (1973). Google Scholar

[5] 5. Faith, C., Algebra II: ring theory (Springer-Verlag, Berlin, Heidelberg, New York, 1976). Google Scholar

[6] 6. Heinicke, A. G., On the ring of quotients at a prime ideal of a right Noetherian ring, Can. J. Math. 24 (1972), 703–712. Google Scholar

[7] 7. Jategaonkar, A. V., Relative Krull dimension and prime ideals in right Noetherian rings, Comm. Algebra 2 (1974), 429–468. Google Scholar

[8] 8. Jategaonkar, A. V., Infective modules and localization in non-commutative Noetherian rings, Trans. A.M.S. 190 (1974), 109–123. Google Scholar

[9] 9. Jategaonkar, A. V., Certain infectives are Artinian, in Non-commutative ring theory, Lecture Notes in Mathematics 545 (Springer-Verlag, Berlin, Heidelberg, New York, 1976). Google Scholar

[10] 10. Lambek, J. and Michler, G., The torsion theory at a prime ideal of a right Noetherian ring, J. Algebra 25 (1973), 364–389. Google Scholar

[11] 11. Lambek, J. and Michler, G., Localization of right Noetherian rings at semiprime ideals, Can. J. Math. 26 (1974), 1069–1085. Google Scholar

[12] 12. Ludgate, A. T., A note on non-commutative Noetherian rings, J. London Math. Soc. 5 (1972), 406–408. Google Scholar

[13] 13. McConnell, J. C., Localization in enveloping rings, J. London Math Soc. Ifi (1968), 421-428: erratum and addendum, J. London Math Soc. 3 (1971), 409–410. Google Scholar

[14] 14. McConnell, J. C., The Noetherian property in complete rings and modules, J. Algebra 12 (1969), 143–153. Google Scholar

[15] 15. McConnell, J. C., On completions of non-commutative Noetherian rings, Comm. Algebra 6 (1978), 1485–1488. Google Scholar

[16] 16. Mueller, B. J., Linear compactness and Morita duality, J. Algebra 16 (1970), 60–66. Google Scholar

[17] 17. Mueller, B. J., Localization in non-commutative Noetherian rings, Can. J. Math. 28 (1976), 600–610. Google Scholar

[18] 18. Passman, D. S., The algebraic structure of group rings (John Wiley and Sons, New York, 1977). Google Scholar

[19] 19. Rentschler, R. and Gabriel, P., Sur la dimension des anneaux et ensembles ordonnes, C. R. Acad. Sci. Paris 265 (1967), 712–715. Google Scholar

[20] 20. Rosenberg, A. and Zelinsky, D., On the finiteness of the infective hull, Math. Z. 70 (1959), 327–380. Google Scholar

[21] 21. Smith, P. F., Localization and the Artin-Rees property, Proc. London Math. Soc. 22 (1971), 39–68. Google Scholar

[22] 22. Smith, P. F., Onnon-commutative regular local rings, Glasgow Math. J. 17 (1976), 98–102. Google Scholar

[23] 23. Walker, R., Local rings and normalizing sets of elements, Proc. London Math. Soc. 24 (1972), 27–45. Google Scholar

[24] 24. Varnos, P., Semi-local Noetherian Wrings, Bull. London Math Soc. 9 (1977), 251–256. Google Scholar

Cité par Sources :