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FINITE LATTICES OF PRERADICALS AND FINITE REPRESENTATION TYPE RINGS

Year 2017, Volume 21, Issue 21, 103 - 120, 17.01.2017
https://doi.org/10.24330/ieja.296155

Abstract

In this paper we study some classes of rings which have a finite lattice of preradicals. We characterize commutative rings with this condition as finite representation type rings, i.e., artinian principal ideal rings. In general, it is easy to see that the lattice of preradicals of a left pure semisimple ring is a set, but it may be infinite. In fact, for a finite dimensional path algebra Λ over an algebraically closed field we prove that Λ-pr is finite if and only if its quiver is a disjoint union of finite quivers of type An; hence there are path algebras of finite representation type such that its lattice of preradicals is an infinite set. As an example, we describe the lattice of preradicals over Λ = kQ when Q is of type An and it has the canonical orientation

References

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  • Texts, 65, Cambridge University Press, Cambridge, 2006.
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  • (1974), 269–310.
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  • Cambridge Studies in Advanced Mathematics, 36, Cambridge University Press, Cambridge, 1997.
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  • Notes in Pure and Applied Mathematics, 75, Marcel Dekker, Inc., New York,1982.
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  • Society, New York, 1948.
  • [
  • [6] D. Eisenbud and P. Griffith, The structure of serial rings, Pacific J. Math., 36(1971), 109–121.
  • [7] R. Fern´andez-Alonso and S. Gavito, The lattice of preradicals over local uniserial
  • rings, J. Algebra Appl., 5(6) (2006), 731–746.
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  • [9] G. Gr¨atzer, General Lattice Theory, Birkh¨auser Verlag, Basel, 2003.
  • [10] L. Gruson and C. U. Jensen, Deux applications de la notion de L-dimension,
  • C. R. Acad. Sci. Paris, S´er. A-B, 282(1) (1976), 23–24.
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  • and representation type, Infinite length modules, Trends Math.,(2000), 331–367.
  • [12] F. Raggi, J. R. Montes, H. Rinc´on, R. Fern´andez-Alonso and C. Signoret, The
  • lattice structure of preradicals, Comm. Algebra, 30(3) (2002), 1533–1544.
  • [13] F. Raggi, J. R´ıos, H. Rinc´on, R. Fern´andez-Alonso and C. Signoret, The lattice
  • structure of preradicals II: Partitions, J. Algebra Appl., 1(2) (2002), 201–214.
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  • Advanced Books and Software, Monterey, CA, 1986.
  • [15] B. Stenstr¨om, Rings of Quotients, Die Grundlehrem der Mathematischen Wissenschaften
  • 217, Springer-Verlag, New York-Heidelberg, 1975.
  • [16] B. Zimmermann-Huisgen and W. Zimmermann, On the sparsity of representations
  • of rings of pure global dimension zero, Trans. Amer. Math. Soc., 320(2)
  • (1990), 695–711.

Year 2017, Volume 21, Issue 21, 103 - 120, 17.01.2017
https://doi.org/10.24330/ieja.296155

Abstract

References

  • [1] I. Assem, D. Simson and A. Skowronski, Elements of the Representation Theory
  • of Associative Algebras, Vol.1, London Mathematical Society Student
  • Texts, 65, Cambridge University Press, Cambridge, 2006.
  • [2] M. Auslander, Representation theory of Artin algebras II, Comm. Algebra, 1
  • (1974), 269–310.
  • [3] M. Auslander, I. Reiten and S. O. Smalø, Representation Theory of Artin Algebras,
  • Cambridge Studies in Advanced Mathematics, 36, Cambridge University Press, Cambridge, 1997.
  • [4] L. Bican, T. Kepka and P. Nemec, Rings, Modules and Preradicals, Lecture
  • Notes in Pure and Applied Mathematics, 75, Marcel Dekker, Inc., New York,1982.
  • [5] G. Birkhoff, Lattice Theory, Colloquium Publications XXV, American Mathematical
  • Society, New York, 1948.
  • [
  • [6] D. Eisenbud and P. Griffith, The structure of serial rings, Pacific J. Math., 36(1971), 109–121.
  • [7] R. Fern´andez-Alonso and S. Gavito, The lattice of preradicals over local uniserial
  • rings, J. Algebra Appl., 5(6) (2006), 731–746.
  • [8] P. Gabriel, Unzerlegbare Darstellungen I, Manuscripta Math., 6 (1972), 71–103.
  • [9] G. Gr¨atzer, General Lattice Theory, Birkh¨auser Verlag, Basel, 2003.
  • [10] L. Gruson and C. U. Jensen, Deux applications de la notion de L-dimension,
  • C. R. Acad. Sci. Paris, S´er. A-B, 282(1) (1976), 23–24.
  • [11] B. Huisgen-Zimmermann, Purity, algebraic compactness, direct sum decompositions
  • and representation type, Infinite length modules, Trends Math.,(2000), 331–367.
  • [12] F. Raggi, J. R. Montes, H. Rinc´on, R. Fern´andez-Alonso and C. Signoret, The
  • lattice structure of preradicals, Comm. Algebra, 30(3) (2002), 1533–1544.
  • [13] F. Raggi, J. R´ıos, H. Rinc´on, R. Fern´andez-Alonso and C. Signoret, The lattice
  • structure of preradicals II: Partitions, J. Algebra Appl., 1(2) (2002), 201–214.
  • [14] R. P. Stanley, Enumerative Combinatorics, Vol. I, Wadsworth and Brooks/Cole
  • Advanced Books and Software, Monterey, CA, 1986.
  • [15] B. Stenstr¨om, Rings of Quotients, Die Grundlehrem der Mathematischen Wissenschaften
  • 217, Springer-Verlag, New York-Heidelberg, 1975.
  • [16] B. Zimmermann-Huisgen and W. Zimmermann, On the sparsity of representations
  • of rings of pure global dimension zero, Trans. Amer. Math. Soc., 320(2)
  • (1990), 695–711.

Details

Journal Section Articles
Authors

Rogelio Fernandez-Alonso This is me


Dolors Herbera>

Publication Date January 17, 2017
Published in Issue Year 2017, Volume 21, Issue 21

Cite

Bibtex @research article { ieja296155, journal = {International Electronic Journal of Algebra}, issn = {1306-6048}, eissn = {1306-6048}, address = {1710 Sokak, No:41, Batikent/Ankara}, publisher = {Abdullah HARMANCI}, year = {2017}, volume = {21}, number = {21}, pages = {103 - 120}, doi = {10.24330/ieja.296155}, title = {FINITE LATTICES OF PRERADICALS AND FINITE REPRESENTATION TYPE RINGS}, key = {cite}, author = {Fernandez-alonso, Rogelio and Herbera, Dolors} }
APA Fernandez-alonso, R. & Herbera, D. (2017). FINITE LATTICES OF PRERADICALS AND FINITE REPRESENTATION TYPE RINGS . International Electronic Journal of Algebra , 21 (21) , 103-120 . DOI: 10.24330/ieja.296155
MLA Fernandez-alonso, R. , Herbera, D. "FINITE LATTICES OF PRERADICALS AND FINITE REPRESENTATION TYPE RINGS" . International Electronic Journal of Algebra 21 (2017 ): 103-120 <https://dergipark.org.tr/en/pub/ieja/issue/27921/296155>
Chicago Fernandez-alonso, R. , Herbera, D. "FINITE LATTICES OF PRERADICALS AND FINITE REPRESENTATION TYPE RINGS". International Electronic Journal of Algebra 21 (2017 ): 103-120
RIS TY - JOUR T1 - FINITE LATTICES OF PRERADICALS AND FINITE REPRESENTATION TYPE RINGS AU - RogelioFernandez-alonso, DolorsHerbera Y1 - 2017 PY - 2017 N1 - doi: 10.24330/ieja.296155 DO - 10.24330/ieja.296155 T2 - International Electronic Journal of Algebra JF - Journal JO - JOR SP - 103 EP - 120 VL - 21 IS - 21 SN - 1306-6048-1306-6048 M3 - doi: 10.24330/ieja.296155 UR - https://doi.org/10.24330/ieja.296155 Y2 - 2016 ER -
EndNote %0 International Electronic Journal of Algebra FINITE LATTICES OF PRERADICALS AND FINITE REPRESENTATION TYPE RINGS %A Rogelio Fernandez-alonso , Dolors Herbera %T FINITE LATTICES OF PRERADICALS AND FINITE REPRESENTATION TYPE RINGS %D 2017 %J International Electronic Journal of Algebra %P 1306-6048-1306-6048 %V 21 %N 21 %R doi: 10.24330/ieja.296155 %U 10.24330/ieja.296155
ISNAD Fernandez-alonso, Rogelio , Herbera, Dolors . "FINITE LATTICES OF PRERADICALS AND FINITE REPRESENTATION TYPE RINGS". International Electronic Journal of Algebra 21 / 21 (January 2017): 103-120 . https://doi.org/10.24330/ieja.296155
AMA Fernandez-alonso R. , Herbera D. FINITE LATTICES OF PRERADICALS AND FINITE REPRESENTATION TYPE RINGS. IEJA. 2017; 21(21): 103-120.
Vancouver Fernandez-alonso R. , Herbera D. FINITE LATTICES OF PRERADICALS AND FINITE REPRESENTATION TYPE RINGS. International Electronic Journal of Algebra. 2017; 21(21): 103-120.
IEEE R. Fernandez-alonso and D. Herbera , "FINITE LATTICES OF PRERADICALS AND FINITE REPRESENTATION TYPE RINGS", International Electronic Journal of Algebra, vol. 21, no. 21, pp. 103-120, Jan. 2017, doi:10.24330/ieja.296155