Relatively Noetherian rings, localization and sheaves. I: The relative second layer condition. (English) Zbl 0834.16032
The authors introduce a relative version of the (strong) second layer condition for rings \(R\) which are relatively Noetherian with respect to some radical in \(R\)-mod and study its impact on the Artin-Rees property. The paper also contains a study of a relative version of the (weak) Artin-Rees property for the ring \(R\) and its characterizations in various ways using the notions of biradicals, stable radicals, and link closedness.
Reviewer: Y.Kurata (Hiratsuka)
MSC:
| 16S90 | Torsion theories; radicals on module categories (associative algebraic aspects) |
| 16D25 | Ideals in associative algebras |
| 16P50 | Localization and associative Noetherian rings |
| 18E15 | Grothendieck categories (MSC2010) |
| 18E35 | Localization of categories, calculus of fractions |
| 14A22 | Noncommutative algebraic geometry |
| 18F20 | Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) |
Keywords:
relative Noetherian rings; second layer condition; Artin-Rees property; biradicals; stable radicals; link closednessReferences:
| [1] | Asensio, M. J. and Torrecillas, B., A note on relative FBN rings,Extracta Math. 4 (1989), 24-26. |
| [2] | Beachy, J., Stable torsion radicals over F.B.N. rings,J. Pure Appl. Algebra,24 (1982), 235-244. · Zbl 0489.16019 · doi:10.1016/0022-4049(82)90041-X |
| [3] | Bell, A. D., Notes on localization in noncommutative Noetherian Rings, Cuadernos de Algebra 9, Universidad de Granada, Granada, 1988. |
| [4] | Bueso, J. L., Jara, P., and Verschoren, A., Compatibility, stability and sheaves: un ménage à trois, monograph, to appear. · Zbl 0815.16001 |
| [5] | Bueso, J. L., Segura, M. I., and Verschoren, A., Strong stability and sheaves,Comm. Algebra 19 (1991), 2531-2545. · Zbl 0734.16012 · doi:10.1080/00927879108824277 |
| [6] | Gabriel, P., Des catégories abéliennes,Bull. Soc. Math. France 90 (1962), 323-448. |
| [7] | Gabriel, P. and Popescu, N., Caractérisation des catégories abéliennes avec générateurs et limites inductives exactes,C. R. Acad. Sci. Paris,258 (1964), 4188-4190. |
| [8] | Golan, J.,Structure Sheaves over a Noncommutative Ring, M. Dekker, New York, 1980. · Zbl 0436.16021 |
| [9] | Golan, J.,Torsion Theories, Longman, London, 1986. |
| [10] | Goldman, O., Rings and modules of quotients,J. Algebra 13 (1969), 10-47. · Zbl 0201.04002 · doi:10.1016/0021-8693(69)90004-0 |
| [11] | Goodearl, K. and Warfield, R.,An Introduction to Noncommutative Noetherian Rings, Cambridge University Press, Cambridge, 1989. · Zbl 0679.16001 |
| [12] | Hartshorne, R.,Algebraic Geometry, Springer-Verlag, Berlin, 1978. |
| [13] | Jategaonkar, A. V., Injective modules and classical localization in Noetherian Rings,Bull. Amer. Math. Soc. 79 (1973), 152-157. · Zbl 0259.16001 · doi:10.1090/S0002-9904-1973-13136-2 |
| [14] | Jategaonkar, A. V.,Localization in Noetherian Rings, London Math. Soc. Lecture Notes, vol. 98, Cambridge Univ. Press, London, 1986. · Zbl 0589.16014 |
| [15] | Lambek, J. and Michler, G., The torision theory at a prime ideal of a right Noetherian Ring,J. Algebra 25 (1973), 364-389. · Zbl 0259.16018 · doi:10.1016/0021-8693(73)90051-3 |
| [16] | Louden, K., Stable torsion and the spectrum of an FBN ring,J. Pure Appl. Algebra 17 (1979), 173-180. · Zbl 0411.16001 · doi:10.1016/0022-4049(79)90031-8 |
| [17] | McConnell, J. and Robson, J. C.,Noncommutative Noetherian Rings, Wiley Interscience, Chichester, 1988. · Zbl 0693.16004 |
| [18] | Merino, L. and Verschoren, A., Symmetry and localization,Bull. Soc. Math. Belg. 43 (1991), 99-112. · Zbl 0753.16013 |
| [19] | Merino, L. and Verschoren, A., Strongly normalizing extensions,J. Pure Appl. Algebra, accepted. · Zbl 0801.16026 |
| [20] | Merino, L., Localizacion y extensiones de anillos noetherianos, PhD thesis, Granada, 1991. |
| [21] | Mulet, J. and Verschoren, A., On compatibility II,Comm. Algebra,20 (1992), 1897-1905. · Zbl 0811.16026 · doi:10.1080/00927879208824438 |
| [22] | Mulet, J. and Verschoren, A., Strong biradicals and sheaves,Proc. V-iéme Contact Franco-Belge en Algèbre, Mulhouse, 1992, to appear. |
| [23] | Robson, J. C., Cyclic and faithful objects in quotient categories with applications to Noetherian simple or Asano Rings, inNoncommutative Ring Theory, Kent State, 1975. |
| [24] | Segura, M. I., Tarazona, D., and Verschoren, A., On compatibility,Comm. Algebra 17 (1989), 677-690. · Zbl 0697.16027 · doi:10.1080/00927878908823751 |
| [25] | Stenström, B.,Rings of Quotients, Springer-Verlag, Berlin, 1975. · Zbl 0296.16001 |
| [26] | Van Oystaeyen, F.,Prime Spectra in Non-commutative Algebra, Lecture Notes in Math. 444, Springer-Verlag, Berlin, 1975. · Zbl 0302.16001 |
| [27] | Van Oystaeyen, F., Compatibility of kernel functors and localization functors,Bull. Soc. Math. Belg. 28 (1976), 131-137. · Zbl 0424.16004 |
| [28] | Van Oystaeyen, F. and Verschoren, A.,Noncommutative Algebraic Geometry, Lecture Notes in Mathematics 887, Springer-Verlag, Berlin, 1981. · Zbl 0455.16006 |
| [29] | Van Oystaeyen, F. and Verschoren, A.,Relative Invariants of Rings: The Commutative Theory, M. Dekker, New York, 1984. · Zbl 0555.16001 |
| [30] | Verschoren, A., Fully bounded Grothendieck categories, Part I: Locally Noetherian categories,J. Pure Appl. Algebra 21 (1981) 99-122. · Zbl 0469.18005 · doi:10.1016/0022-4049(81)90077-3 |
| [31] | Verschoren, A.,Compatibility and Stability, Notas de Matematica, vol. 3 Murcia, 1990. · Zbl 0708.16008 |
| [32] | Verschoren, A. and Vidal, C., Relatively Noetherian Rings, localization and sheaves. Part II: Structure sheaves,K-Theory 8 (1994), 133-152 (this issue). · Zbl 0834.16033 · doi:10.1007/BF00961454 |
| [33] | Vidal, C., Localización en Categorías localmente noetherianas, PhD. thesis, Universidad de Santiago, 1993. |
| [34] | Vidal, C. and Verschoren, A., Reflecting torsion,Bull. Soc. Math. Belg. 45 (1993), 281-299. · Zbl 0809.16038 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
