Standard prime ideals and lying over for finite extensions of Noetherian algebras

Krause, Günter

Glasgow mathematical journal, Tome 37 (1995) no. 3, pp. 311-326

Résumé

Let k be a field, let R be a noetherian k-algebra of finite Gelfand-Kirillov dimension GK(R), and let M be a finitely generated right R-module. A standard prime factor series for M is a finite sequence of submodules 0 = N0 ⊂ N1 ⊂...⊂ Ni−1 ⊂ Ni ⊂.... ⊂ Nn = M, such that for each i the annihilator Pi = rR (Ni/Ni−1) is the unique associated prime of Ni/Ni−1 and GK(R/Pi)≤ GK(R/Pj) whenever i≤ j. The set of prime ideals arising from such a series is an invariant of M, called the set of standard primes St(M) of M. The concept, inspired by the notion of a standard affiliated series introduced by Lenagan and Warfield in [7], has been developed in [5], where it was shown that St(M) coincides with the set of all those prime ideals that are minimal over the annihilator of a nonzero submodule of M.

Détail

DOI : 10.1017/S0017089500031591

Bibliographie

[1] 1.Goodearl, K. R. and Letzter, E. S., Prime ideals in skew and q-skew polynomial rings, Mem. Amer. Math. Soc, No 521 (1994). Google Scholar

[2] 2.Goodearl, K. R. and Warfield, R. B. Jr, An introduction to noncommulative Noetherian rings, London Mathematical Society Student Texts 16 (Cambridge University Press, 1989). Google Scholar

[3] 3.Hodges, T. J. and Osterlurg, J., A rank two indecomposable projective module over a Noetherian domain of Krull dimension one, Bull. London Math. Soc. 19 (1987), 139–144. Google Scholar | DOI

[4] 4.Krause, G., Additive rank functions in Noetherian rings, J. Algebra 130 (1990), 451–461. Google Scholar | DOI

[5] 5.Krause, G., Prime factor series of modules over a Noetherian algebra, Abelian groups and noncommulative rings, Contemp. Math. 130 (Amer. Math. Soc., 1992), 215–229. Google Scholar

[6] 6.Krause, G. R. and Lenagan, T. H., Growth of algebras and Gelfand-Kirillov dimension, Research Notes in Mathematics, 116 (Pitman, 1985). Google Scholar

[7] 7.Lenagan, T. H. and Warfield, R. B. Jr, Affiliated series and extensions of modules, J. Algebra 142 (1991), 164–187. Google Scholar | DOI

[8] 8.Letzter, E. S., Primitive ideals in finite extensions of Noetherian rings, J. London Math. Soc. (2) 39 (1989), 427–435. Google Scholar | DOI

[9] 9.Letzter, E. S., Prime ideals in finite extensions of Noetherian rings, J. Algebra 135 (1990), 412–438. Google Scholar | DOI

[10] 10.Letzter, E. S., Finite correspondence of spectra in Noetherian ring extensions, Proc. Amer. Math. Soc. 116 (1992), 645–652. Google Scholar | DOI

[11] 11.Lorenz, M., K of skew group rings and simple Noetherian rings without idempotents, J. London Math. Soc. (2) 32 (1985), 41–50. Google Scholar | DOI

[12] 12.Rowen, L. H., Ring theory, Vol. 1 (Academic Press, 1988). Google Scholar

[13] 13.Small, L. W., Stafford, J. T. and Warfield, R. B. Jr, Affine algebras of Gelfand-Kirillov dimension one are PI, Math. Proc. Cambridge Philos. Soc. 97 (1985), 407–414. Google Scholar | DOI

[14] 14.Small, L. W. and Warfield, R. B. Jr, Finite extensions of rings II, to appear. Google Scholar

[15] 15.Warfield, R. B. Jr, Noetherian rings with trace conditions, Trans. Amer. Math. Soc. 331 (1992), 449–463. Google Scholar | DOI