DOI QR코드

DOI QR Code

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

  • Alahmadi, Adel (Department of Mathematics Faculty of Science King Abdulaziz University) ;
  • Facchini, Alberto (Dipartimento di Matematica Universita di Padova)
  • Received : 2012.07.02
  • Published : 2013.05.01

Abstract

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

Keywords

References

  1. A. Alahmadi and S. K. Jain, A note on almost injective modules, Math. J. Okayama Univ. 51 (2009), 101-109.
  2. B. Amini, A. Amini, and A. Facchini, Equivalence of diagonal matrices over local rings, J. Algebra 320 (2008), no. 3, 1288-1310. https://doi.org/10.1016/j.jalgebra.2008.04.008
  3. A. Facchini, Module Theory, Endomorphism rings and direct sum decompositions in some classes of modules, Birkhauser Verlag, Basel, 1998.
  4. A. Facchini, Representations of additive categories and direct-sum decompositions of objects, Indiana Univ. Math. J. 56 (2007), no. 2, 659-680. https://doi.org/10.1512/iumj.2007.56.2865
  5. A. Facchini, A characterization of additive categories with the Krull-Schmidt property, Algebra and its applications, 125-129, Contemp. Math., 419, Amer. Math. Soc., Providence, RI, 2006.
  6. A. Facchini and M. Perone, Maximal ideals in preadditive categories and semilocal categories, J. Algebra Appl. 10 (2011), no. 1, 1-27. https://doi.org/10.1142/S0219498811004458
  7. A. Facchini, On some noteworthy pairs of ideals in Mod-R, to appear in Appl. Categ. Structures, 2012.
  8. A. Facchini and P. Prihoda, Endomorphism rings with finitely many maximal right ideals, Comm. Algebra 39 (2011), no. 9, 3317-3338. https://doi.org/10.1080/00927872.2011.555850
  9. A. Facchini, The Krull-Schmidt Theorem in the case two, Algebr. Represent. Theory 14 (2011), no. 3, 545-570. https://doi.org/10.1007/s10468-009-9202-1
  10. K. R. Goodearl and R. B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, Second Edition, London Math. Soc. Lecture Note Series 61, Cambridge Univ. Press, Cambridge, 2004.
  11. S. H. Mohamed and B. J. Muller, Continuous and Discrete Modules, London Math. Soc. Lecture Note Series 147, Cambdridge Univ. Press, Cambridge, 1990.

Cited by

  1. Direct products of modules whose endomorphism rings have at most two maximal ideals vol.435, 2015, https://doi.org/10.1016/j.jalgebra.2015.04.002
  2. On a category of chains of modules whose endomorphism rings have at most 2n maximal ideals 2018, https://doi.org/10.1080/00927872.2017.1372459