SOME REMARKS ON CATEGORIES OF MODULES MODULO MORPHISMS WITH ESSENTIAL KERNEL OR SUPERFLUOUS IMAGE

Title & Authors
SOME REMARKS ON CATEGORIES OF MODULES MODULO MORPHISMS WITH ESSENTIAL KERNEL OR SUPERFLUOUS IMAGE

Abstract
For an ideal $\small{\mathcal{I}}$ of a preadditive category $\small{\mathcal{A}}$, we study when the canonical functor $\small{\mathcal{C}:\mathcal{A}{\rightarrow}\mathcal{A}/\mathcal{I}}$ is local. We prove that there exists a largest full subcategory $\small{\mathcal{C}}$ of $\small{\mathcal{A}}$, for which the canonical functor $\small{\mathcal{C}:\mathcal{C}{\rightarrow}\mathcal{C}/\mathcal{I}}$ is local. Under this condition, the functor $\small{\mathcal{C}}$, turns out to be a weak equivalence between $\small{\mathcal{C}}$, and $\small{\mathcal{C}/\mathcal{I}}$. If $\small{\mathcal{A}}$ is additive (with splitting idempotents), then $\small{\mathcal{C}}$ is additive (with splitting idempotents). The category $\small{\mathcal{C}}$ is ample in several cases, such as the case when $\small{\mathcal{A}}$=Mod-R and $\small{\mathcal{I}}$ is the ideal $\small{{\Delta}}$ of all morphisms with essential kernel. In this case, the category $\small{\mathcal{C}}$ contains, for instance, the full subcategory $\small{\mathcal{F}}$ of Mod-R whose objects are all the continuous modules. The advantage in passing from the category $\small{\mathcal{F}}$ to the category $\small{\mathcal{F}/\mathcal{I}}$ lies in the fact that, although the two categories $\small{\mathcal{F}}$ and $\small{\mathcal{F}/\mathcal{I}}$ are weakly equivalent, every endomorphism has a kernel and a cokernel in $\small{\mathcal{F}/{\Delta}}$, which is not true in $\small{\mathcal{F}}$. In the final section, we extend our theory from the case of one ideal$\small{\mathcal{I}}$ to the case of $\small{n}$ ideals $\small{\mathcal{I}_}$, $\small{{\ldots}}$, $\small{\mathca{l}_n}$.
Keywords
Language
English
Cited by
1.
Direct products of modules whose endomorphism rings have at most two maximal ideals, Journal of Algebra, 2015, 435, 204
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