Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2013; 50(3): 557-578

Printed May 1, 2013

https://doi.org/10.4134/JKMS.2013.50.3.557

Copyright © The Korean Mathematical Society.

Some remarks on categories of modules modulo morphisms with essential kernel or superfluous image

Adel Alahmadi and Alberto Facchini

King Abdulaziz University, Universit\`a di Padova

Abstract

For an ideal $\Cal I$ of a preadditive category $\Cal A$, we study when the canonical functor $C\colon \Cal A\to\Cal A/\Cal I$ is local. We prove that there exists a largest full subcategory $\Cal C$ of $\Cal A$ for which the canonical functor $C\colon\Cal C\to\Cal C/\Cal I$ is local. Under this condition, the functor $C$ turns out to be a weak equivalence between $\Cal C$ and $\Cal C/\Cal I$. If $\Cal A$ is additive (with splitting idempotents), then $\Cal C$ is additive (with splitting idempotents). The category $\Cal C$ is ample in several cases, such as the case when $\Cal A=\Mod R$ and $\Cal I$ is the ideal $\Delta$ of all morphisms with essential kernel. In this case, the category $\Cal C$ contains, for instance, the full subcategory $\Cal F$ of $\Mod R$ whose objects are all the continuous modules. The advantage in passing from the category $\Cal F$ to the category $\Cal F/\Cal I$ lies in the fact that, although the two categories $\Cal F$ and $\Cal F/\Cal I$ are weakly equivalent, every endomorphism has a kernel and a cokernel in $\Cal F/\Delta$, which is not true in $\Cal F$. In the final section, we extend our theory from the case of one ideal $\Cal I$ to the case of $n$ ideals $\Cal I_1,\dots,\Cal I_n$.

Keywords: preadditive category, additive functor, local functor, essential kernel, superfluous image

MSC numbers: 16D90

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