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CONSTRUCTION OF MODULES WITH A PRESCRIBED DIRECT SUM DECOMPOSITION

Year 2014, , 218 - 248, 01.06.2014
https://doi.org/10.24330/ieja.266249

Abstract

We give some criteria for recognizing local rings that allow us to
show that indecomposable AB5∗ modules over commutative rings and couniform
modules over noetherian commutative rings have a local endomorphism
ring. We also develop some theory on methods to construct modules with
a prescribed direct-sum decomposition. As an application we realize an interesting
class of commutative monoids as monoids of direct summands of a
direct sum of a countable number of copies of a suitable artinian cyclic module,
showing that there may appear a rich supply of direct summands that are
not a direct sum of artinian modules. An important gadget for proving our
realization result is a variation of a method for realizing a given ring as the
endomorphism ring of a cyclic (artinian) module due to Armendariz, Fisher
and Snider.

References

  • E. P. Armendariz, J. W. Fisher and R. L. Snider, On injective and surjective endomorphisms of finitely generated modules, Comm. Alg., 7 (1978), 659–672. [2] R. Camps and W. Dicks, On semilocal rings, Israel J. Math., 81 (1993), 203– 211.
  • R. Camps and A. Facchini, The Pr¨ufer rings that are endomorphism rings of Artinian modules, Comm. Algebra, 22 (1994), 3133–3157.
  • R. Camps and P. Menal, Power cancellation for artinian modules, Comm. Algebra, 19 (1991), 2081–2095.
  • A. Facchini, Fiber products and Morita duality for commutative rings, Rend. Sem. Mat. Univ. Padova, 67 (1982), 143–159.
  • A. Facchini, Module Theory. Endomorphism Rings and Direct Sum Decompo- sitions in Some Classes of Modules, Progress in Math. 167, Birkh¨auser Verlag, Basel, 1998.
  • A. Facchini, D. Herbera, Kof a semilocal ring, J. Algebra, 225 (2000), 47–69. [8] A. Facchini, D. Herbera, L. Levy and P. V´amos, Krull-Schmidt fails for Ar- tinian modules, Proc. Amer. Math. Soc., 123 (1995), 3587–3592.
  • A. Facchini and P. V´amos, Injective modules over pullbacks, J. London Math. Soc., (2) 31(3) (1985), 425–438.
  • C. Faith and D. Herbera, Endomorphism rings and tensor products of linearly compact modules, Comm. Algebra, 25(4) (1997), 1215–1255.
  • J. W. Fisher, Nil subrings of endomorphism rings of modules, Proc. Amer. Math. Soc., 34 (1972), 75–78.
  • D. Eisenbud and J. C. Robson, Hereditary Noetherian prime rings, J. of Alge- bra, 16 (1970), 86–104.
  • V. N. Gerasimov and I. I. Sakhaev, A counterexample to two hypotheses on pro- jective and flat modules (Russian), Sib. Mat. Zh., 25(6) (1984), 31–35. English translation: Sib. Math. J., 24 (1984), 855–859.
  • K. R. Goodearl and R. B. Warfield, Simple modules over hereditary Noetherian prime rings, J. of Algebra, 57 (1979), 82–100.
  • K. R. Goodearl and R. B. Warfield, An Introduction to Noncommutative Noe- therian Rings, London Mathematical Society Student Texts 16, Cambridge University Press, Cambridge, 1989.
  • D. Herbera and P. Pˇr´ıhoda, Big projective modules over Noetherian semilocal rings, J. Reine und Angew. Math., 648 (2010), 111–148.
  • D. Herbera and P. Pˇr´ıhoda, Infinitely generated modules over pullbacks of rings, Trans. Amer. Math. Soc., 366(3) (2014), 1433–1454.
  • D. Herbera and A. Shamsuddin, Modules with semi-local endomorphism ring, Proc. Amer. Math. Soc., 123 (1995), 3593–3600.
  • W. Krull, Matrizen, Moduln und verallgemeinerte Abelsche Gruppen im Bere- ich der ganzen algebraischen Zahlen, Heidelberger Akad. Wiss., 2 (1932), 13– 38.
  • B. Lemonnier, AB-5*et la dualit´e de Morita, C.R. Acad. Sc. Paris, 289 (1979), Series A, 47-50.
  • T. H. Lenagan, Bounded hereditary Noetherian prime rings, J. London Math. Soc., 6(2) (1973), 241–246.
  • G. J. Leuschke and R. Wiegand, Cohen-Macaulay Representations, Mathe- matical Surveys and Monographs, 181, American Mathematical Society, Prov- idence, RI, 2012.
  • J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, Grad- uate Studies in Mathematics 30, American Mathematical Society, Providence (Rhode Island), 1987 (reprinted 2001).
  • E. Matlis, 1-Dimensional Cohen-Macaulay rings, Lect. Notes in Math., 327, Springer-Verlag, 1973.
  • K. I. Pimenov and A. V. Yakovlev, Artinian modules over a matrix ring, In: Infinite Lenght Modules, H. Krause and C. M. Ringel Eds., Trends in Math., Birkh¨auser, Basel, 2000, 101–105.
  • P. Pˇr´ıhoda, Projective modules are determined by their radical factors, J. Pure Applied Algebra, 210 (2007), 827–835.
  • G. Puninski, When every projective module is a direct sum of finitely generated modules, preprint 2005.
  • C. M. Ringel, Krull-Remak.Schmidt fails for Artinian modules over local rings, Algebras and Rep. Theory, 4 (2001), 77–86.
  • R. B. Warfield Jr., Exchange rings and decompositions of modules, Math. Ann., 199 (1972), 31–36.
  • R. B. Warfield Jr., Cancellation of modules and groups and stable range of endomorphism rings, Pacific J. Math., 91 (1980), 457–485.
  • R. Wiegand, Direct-sum decompositions over local rings, J. Algebra, 240(1) (2001), 83–97.
  • A. V. Yakovlev, On direct decompositions of p-adic groups, (Russian) Algebra i Analiz, 12(6) (2000), 217–223; translation in St. Petersburg Math. J., 12(6) (2001), 1043–1047.
  • B. Zimmermann-Huisgen and W. Zimmermann, Classes of modules with the exchange property, J. Algebra, 88(2) (1984), 416–434.
  • W. Zimmermann, Rein injektive direkte summen von moduln, Comm. Algebra, 5(10) (1977), 1083–1117. Dolors Herbera
  • Departament de Matem`atiques
  • Universitat Aut`onoma de Barcelona
  • E-08193 Bellaterra (Barcelona), Spain
  • e-mail: dolors@mat.uab.cat
Year 2014, , 218 - 248, 01.06.2014
https://doi.org/10.24330/ieja.266249

Abstract

References

  • E. P. Armendariz, J. W. Fisher and R. L. Snider, On injective and surjective endomorphisms of finitely generated modules, Comm. Alg., 7 (1978), 659–672. [2] R. Camps and W. Dicks, On semilocal rings, Israel J. Math., 81 (1993), 203– 211.
  • R. Camps and A. Facchini, The Pr¨ufer rings that are endomorphism rings of Artinian modules, Comm. Algebra, 22 (1994), 3133–3157.
  • R. Camps and P. Menal, Power cancellation for artinian modules, Comm. Algebra, 19 (1991), 2081–2095.
  • A. Facchini, Fiber products and Morita duality for commutative rings, Rend. Sem. Mat. Univ. Padova, 67 (1982), 143–159.
  • A. Facchini, Module Theory. Endomorphism Rings and Direct Sum Decompo- sitions in Some Classes of Modules, Progress in Math. 167, Birkh¨auser Verlag, Basel, 1998.
  • A. Facchini, D. Herbera, Kof a semilocal ring, J. Algebra, 225 (2000), 47–69. [8] A. Facchini, D. Herbera, L. Levy and P. V´amos, Krull-Schmidt fails for Ar- tinian modules, Proc. Amer. Math. Soc., 123 (1995), 3587–3592.
  • A. Facchini and P. V´amos, Injective modules over pullbacks, J. London Math. Soc., (2) 31(3) (1985), 425–438.
  • C. Faith and D. Herbera, Endomorphism rings and tensor products of linearly compact modules, Comm. Algebra, 25(4) (1997), 1215–1255.
  • J. W. Fisher, Nil subrings of endomorphism rings of modules, Proc. Amer. Math. Soc., 34 (1972), 75–78.
  • D. Eisenbud and J. C. Robson, Hereditary Noetherian prime rings, J. of Alge- bra, 16 (1970), 86–104.
  • V. N. Gerasimov and I. I. Sakhaev, A counterexample to two hypotheses on pro- jective and flat modules (Russian), Sib. Mat. Zh., 25(6) (1984), 31–35. English translation: Sib. Math. J., 24 (1984), 855–859.
  • K. R. Goodearl and R. B. Warfield, Simple modules over hereditary Noetherian prime rings, J. of Algebra, 57 (1979), 82–100.
  • K. R. Goodearl and R. B. Warfield, An Introduction to Noncommutative Noe- therian Rings, London Mathematical Society Student Texts 16, Cambridge University Press, Cambridge, 1989.
  • D. Herbera and P. Pˇr´ıhoda, Big projective modules over Noetherian semilocal rings, J. Reine und Angew. Math., 648 (2010), 111–148.
  • D. Herbera and P. Pˇr´ıhoda, Infinitely generated modules over pullbacks of rings, Trans. Amer. Math. Soc., 366(3) (2014), 1433–1454.
  • D. Herbera and A. Shamsuddin, Modules with semi-local endomorphism ring, Proc. Amer. Math. Soc., 123 (1995), 3593–3600.
  • W. Krull, Matrizen, Moduln und verallgemeinerte Abelsche Gruppen im Bere- ich der ganzen algebraischen Zahlen, Heidelberger Akad. Wiss., 2 (1932), 13– 38.
  • B. Lemonnier, AB-5*et la dualit´e de Morita, C.R. Acad. Sc. Paris, 289 (1979), Series A, 47-50.
  • T. H. Lenagan, Bounded hereditary Noetherian prime rings, J. London Math. Soc., 6(2) (1973), 241–246.
  • G. J. Leuschke and R. Wiegand, Cohen-Macaulay Representations, Mathe- matical Surveys and Monographs, 181, American Mathematical Society, Prov- idence, RI, 2012.
  • J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, Grad- uate Studies in Mathematics 30, American Mathematical Society, Providence (Rhode Island), 1987 (reprinted 2001).
  • E. Matlis, 1-Dimensional Cohen-Macaulay rings, Lect. Notes in Math., 327, Springer-Verlag, 1973.
  • K. I. Pimenov and A. V. Yakovlev, Artinian modules over a matrix ring, In: Infinite Lenght Modules, H. Krause and C. M. Ringel Eds., Trends in Math., Birkh¨auser, Basel, 2000, 101–105.
  • P. Pˇr´ıhoda, Projective modules are determined by their radical factors, J. Pure Applied Algebra, 210 (2007), 827–835.
  • G. Puninski, When every projective module is a direct sum of finitely generated modules, preprint 2005.
  • C. M. Ringel, Krull-Remak.Schmidt fails for Artinian modules over local rings, Algebras and Rep. Theory, 4 (2001), 77–86.
  • R. B. Warfield Jr., Exchange rings and decompositions of modules, Math. Ann., 199 (1972), 31–36.
  • R. B. Warfield Jr., Cancellation of modules and groups and stable range of endomorphism rings, Pacific J. Math., 91 (1980), 457–485.
  • R. Wiegand, Direct-sum decompositions over local rings, J. Algebra, 240(1) (2001), 83–97.
  • A. V. Yakovlev, On direct decompositions of p-adic groups, (Russian) Algebra i Analiz, 12(6) (2000), 217–223; translation in St. Petersburg Math. J., 12(6) (2001), 1043–1047.
  • B. Zimmermann-Huisgen and W. Zimmermann, Classes of modules with the exchange property, J. Algebra, 88(2) (1984), 416–434.
  • W. Zimmermann, Rein injektive direkte summen von moduln, Comm. Algebra, 5(10) (1977), 1083–1117. Dolors Herbera
  • Departament de Matem`atiques
  • Universitat Aut`onoma de Barcelona
  • E-08193 Bellaterra (Barcelona), Spain
  • e-mail: dolors@mat.uab.cat
There are 36 citations in total.

Details

Other ID JA77TK62SD
Journal Section Articles
Authors

Dolors Herbera This is me

Publication Date June 1, 2014
Published in Issue Year 2014

Cite

APA Herbera, D. (2014). CONSTRUCTION OF MODULES WITH A PRESCRIBED DIRECT SUM DECOMPOSITION. International Electronic Journal of Algebra, 15(15), 218-248. https://doi.org/10.24330/ieja.266249
AMA Herbera D. CONSTRUCTION OF MODULES WITH A PRESCRIBED DIRECT SUM DECOMPOSITION. IEJA. June 2014;15(15):218-248. doi:10.24330/ieja.266249
Chicago Herbera, Dolors. “CONSTRUCTION OF MODULES WITH A PRESCRIBED DIRECT SUM DECOMPOSITION”. International Electronic Journal of Algebra 15, no. 15 (June 2014): 218-48. https://doi.org/10.24330/ieja.266249.
EndNote Herbera D (June 1, 2014) CONSTRUCTION OF MODULES WITH A PRESCRIBED DIRECT SUM DECOMPOSITION. International Electronic Journal of Algebra 15 15 218–248.
IEEE D. Herbera, “CONSTRUCTION OF MODULES WITH A PRESCRIBED DIRECT SUM DECOMPOSITION”, IEJA, vol. 15, no. 15, pp. 218–248, 2014, doi: 10.24330/ieja.266249.
ISNAD Herbera, Dolors. “CONSTRUCTION OF MODULES WITH A PRESCRIBED DIRECT SUM DECOMPOSITION”. International Electronic Journal of Algebra 15/15 (June 2014), 218-248. https://doi.org/10.24330/ieja.266249.
JAMA Herbera D. CONSTRUCTION OF MODULES WITH A PRESCRIBED DIRECT SUM DECOMPOSITION. IEJA. 2014;15:218–248.
MLA Herbera, Dolors. “CONSTRUCTION OF MODULES WITH A PRESCRIBED DIRECT SUM DECOMPOSITION”. International Electronic Journal of Algebra, vol. 15, no. 15, 2014, pp. 218-4, doi:10.24330/ieja.266249.
Vancouver Herbera D. CONSTRUCTION OF MODULES WITH A PRESCRIBED DIRECT SUM DECOMPOSITION. IEJA. 2014;15(15):218-4.