Resolvents and dimensions of modules and rings. (English) Zbl 0694.16012
In this paper, we show that for a left and right noetherian ring R and an integer n, the left and right injective dimensions of R are at most n if and only if projective resolutions of the nth syzygies of all R-modules (left and right) are flat resolvents, and if and only if injective resolutions of the nth cosyzygies of all R-modules (left and right) are injective resolvents. Then we characterize the weak global dimension of left coherent rings in terms of projective resolvents. This shows that the right global dimension of left coherent rings with respect to flat resolvents is determined by the projective resolvents of finitely presented modules. Finally, we characterize the left global dimension of left noetherian rings in terms of injective precovers and projective preenvelopes.
Reviewer: O.M.G.Jenda
MSC:
| 16P40 | Noetherian rings and modules (associative rings and algebras) |
| 16D40 | Free, projective, and flat modules and ideals in associative algebras |
| 16E10 | Homological dimension in associative algebras |
| 16D50 | Injective modules, self-injective associative rings |
Keywords:
left and right noetherian ring; left and right injective dimensions; projective resolutions; flat resolvents; injective resolutions; cosyzygies; injective resolvents; weak global dimension; left coherent rings; right global dimension; finitely presented modulesReferences:
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