Turkish Journal of Mathematics
DOI
10.3906/mat-2107-13
Abstract
We prove a stronger form of an analogue of a Kaplansky lemma on homological dimensions by showing that in a pure-exact sequence $0 \to A \to B \to C \to 0$, the weak dimensions of the modules satisfy $ {\rm w.d.} B = \max\{{\rm w.d.} A, {\rm w.d.} C\}$. We also show that the same equality holds for the injective dimensions whenever the ring is noetherian. In addition, a version of Auslander's lemma for chains of pure submodules $M_{\rho}$ is proved: the weak dimension of the union of the chain equals the supremum of the weak dimensions of the factor modules $M_{\rho+1}/ M_{\rho}$ in the chain. The same holds for injective dimensions if the ring is noetherian.
Keywords
Tor, pure submodules, weak dimension, injective dimension
First Page
1899
Last Page
1902
Recommended Citation
FUCHS, LASZLO and LEE, SANG BUM
(2021)
"Weak and injective dimensional analogues of Kaplansky's and Auslander's lemmas for purity,"
Turkish Journal of Mathematics: Vol. 45:
No.
5, Article 1.
https://doi.org/10.3906/mat-2107-13
Available at:
https://journals.tubitak.gov.tr/math/vol45/iss5/1
