Dimension fa1ble negative globale des anneaux commutatifs

Abstract

Using definitions and properties by E.E. Enochs [1], V. K. Akatsa [2], E.E. Enochs and O.Mg Jenda [3], we prove first, that, if (R _α)_α∈I is a direct system of coherent rings, so that, for every α ∈ I, the limit R is a rα-flat module, then the negative-weak-global dimension of R is bounded by the upper bound of the negative-weak-global dimensions of the rings R _α

Then, if is a product of rings so that every R-module admits a flat envelope, the negative-weak-global dimension of R is bounded by the upper bound of the negative weak-global dimensions of the rings R_i . We study a class of inverse systems of modules and prove that inverse limit of FP-injective modules whose set of index is N and morphisms are onto is a module of negative-weak dimension zero

Finally, using results of L. FUCHS [4] we give examples of rings of which we compute the homogical dimension, the weak-global dimension, the negative-weak-global dimension and the pure-global dimension.

Keywords:

• Coherent rings
• fflat envelopes
• negative-weak-global dimension
• pure-global dimension
• FP-injective dimension