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Year 2021, , 285 - 303, 17.07.2021
https://doi.org/10.24330/ieja.969940

Abstract

References

  • A. Barnard, Distributive extensions of modules, J. Algebra, 70(2) (1981), 303-315.
  • P. M. Cohn, Free Rings and Their Relations, Second Edition, London Mathematical Society Monographs, 19, Academic Press, Inc., London, 1985.
  • T. M. K. Davison, Distributive homomorphisms of rings and modules, J. Reine Angew. Math., 271 (1974), 28-34.
  • J. M. Garcia, P. Jara and L. M. Merino, Decomposition of comodules, Comm. Algebra, 27(4) (1999), 1797-1805.
  • G. Gratzer, General Lattice Theory, Birkhauser Verlag, Basel-Stuttgart, 1978.
  • C. U. Jensen, Arithmetical rings, Acta Math. Acad. Sci. Hungar., 17 (1966), 115-123.
  • S. Rajaee, Multiplication modules on arithmetical rings, Int. J. Algebra, 7 (2013), 825-828.
  • B. Stenstrom, Rings of Quotients, Springer-Verlag, Berlin, 1975.
  • W. Stephenson, Modules whose lattice of submodules is distributive, Proc. London Math. Soc., 28(3) (1974), 291-310.
  • A. A. Tuganbaev, Distributive extensions of modules, J. Math. Sci., 149(3) (2008), 1279-1285.
  • N. Vanaja, All finitely generated M-subgenerated modules are extending, Comm. Algebra, 24(2) (1996), 543-572.
  • R. Wisbauer, Grundlagen der Modul- und Ringtheorie, Verlag Reinhard Fischer, Munich, 1988.
  • R. Wisbauer, Decompositions of modules and comodules, Algebra and its Applications (Athens, OH, 1999), Contemp. Math., Amer. Math. Soc., Providence, RI, 259 (2000), 547-561.

LATTICE DECOMPOSITION OF MODULES

Year 2021, , 285 - 303, 17.07.2021
https://doi.org/10.24330/ieja.969940

Abstract

The first aim of this work is to characterize when the lattice of all submodules of a module is a direct product of two lattices. In particular, which decompositions of a module $M$ produce these decompositions: the \emph{lattice decompositions}. In a first \textit{\'{e}tage} this can be done using endomorphisms of $M$, which produce a decomposition of the ring $\End_R(M)$ as a product of rings, i.e., they are central idempotent endomorphisms. But since not every central idempotent endomorphism produces a lattice decomposition, the classical theory is not of application. In a second step we characterize when a particular module $M$ has a lattice decomposition; this can be done, in the commutative case in a simple way using the support, $\Supp(M)$, of $M$; but, in general, it is not so easy. Once we know when a module decomposes, we look for characterizing its decompositions. We show that a good framework for this study, and its generalizations, could be provided by the category $\sigma[M]$, the smallest Grothendieck subcategory of $\rMod{R}$ containing $M$.

References

  • A. Barnard, Distributive extensions of modules, J. Algebra, 70(2) (1981), 303-315.
  • P. M. Cohn, Free Rings and Their Relations, Second Edition, London Mathematical Society Monographs, 19, Academic Press, Inc., London, 1985.
  • T. M. K. Davison, Distributive homomorphisms of rings and modules, J. Reine Angew. Math., 271 (1974), 28-34.
  • J. M. Garcia, P. Jara and L. M. Merino, Decomposition of comodules, Comm. Algebra, 27(4) (1999), 1797-1805.
  • G. Gratzer, General Lattice Theory, Birkhauser Verlag, Basel-Stuttgart, 1978.
  • C. U. Jensen, Arithmetical rings, Acta Math. Acad. Sci. Hungar., 17 (1966), 115-123.
  • S. Rajaee, Multiplication modules on arithmetical rings, Int. J. Algebra, 7 (2013), 825-828.
  • B. Stenstrom, Rings of Quotients, Springer-Verlag, Berlin, 1975.
  • W. Stephenson, Modules whose lattice of submodules is distributive, Proc. London Math. Soc., 28(3) (1974), 291-310.
  • A. A. Tuganbaev, Distributive extensions of modules, J. Math. Sci., 149(3) (2008), 1279-1285.
  • N. Vanaja, All finitely generated M-subgenerated modules are extending, Comm. Algebra, 24(2) (1996), 543-572.
  • R. Wisbauer, Grundlagen der Modul- und Ringtheorie, Verlag Reinhard Fischer, Munich, 1988.
  • R. Wisbauer, Decompositions of modules and comodules, Algebra and its Applications (Athens, OH, 1999), Contemp. Math., Amer. Math. Soc., Providence, RI, 259 (2000), 547-561.
There are 13 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

J. M. Garcıa This is me

P. Jara This is me

L. M. Merıno This is me

Publication Date July 17, 2021
Published in Issue Year 2021

Cite

APA Garcıa, J. M., Jara, P., & Merıno, L. M. (2021). LATTICE DECOMPOSITION OF MODULES. International Electronic Journal of Algebra, 30(30), 285-303. https://doi.org/10.24330/ieja.969940
AMA Garcıa JM, Jara P, Merıno LM. LATTICE DECOMPOSITION OF MODULES. IEJA. July 2021;30(30):285-303. doi:10.24330/ieja.969940
Chicago Garcıa, J. M., P. Jara, and L. M. Merıno. “LATTICE DECOMPOSITION OF MODULES”. International Electronic Journal of Algebra 30, no. 30 (July 2021): 285-303. https://doi.org/10.24330/ieja.969940.
EndNote Garcıa JM, Jara P, Merıno LM (July 1, 2021) LATTICE DECOMPOSITION OF MODULES. International Electronic Journal of Algebra 30 30 285–303.
IEEE J. M. Garcıa, P. Jara, and L. M. Merıno, “LATTICE DECOMPOSITION OF MODULES”, IEJA, vol. 30, no. 30, pp. 285–303, 2021, doi: 10.24330/ieja.969940.
ISNAD Garcıa, J. M. et al. “LATTICE DECOMPOSITION OF MODULES”. International Electronic Journal of Algebra 30/30 (July 2021), 285-303. https://doi.org/10.24330/ieja.969940.
JAMA Garcıa JM, Jara P, Merıno LM. LATTICE DECOMPOSITION OF MODULES. IEJA. 2021;30:285–303.
MLA Garcıa, J. M. et al. “LATTICE DECOMPOSITION OF MODULES”. International Electronic Journal of Algebra, vol. 30, no. 30, 2021, pp. 285-03, doi:10.24330/ieja.969940.
Vancouver Garcıa JM, Jara P, Merıno LM. LATTICE DECOMPOSITION OF MODULES. IEJA. 2021;30(30):285-303.