1. DergiPark
  2. International Electronic Journal of Algebra
  3. Arşiv
  4. Cilt 15, Sayı 15
Yıl 2014, Cilt 15, Sayı 15, Sayfalar 218 - 248 2014-06-01

CONSTRUCTION OF MODULES WITH A PRESCRIBED DIRECT SUM DECOMPOSITION

Dolors Herbera [1]

183 220

Öz

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.

Anahtar Kelimeler

AB5∗-module, artinian module, semilocal ring, couniform module, category equivalence, monoid, direct-sum, pullback, pushout

Kaynakça

  • 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

Ayrıntılar

Konular
Diğer ID JA77TK62SD
Dergi Bölümü Makaleler
Yazarlar

Yazar: Dolors Herbera

Kaynak Göster

Bibtex @ { ieja266249, journal = {International Electronic Journal of Algebra}, issn = {1306-6048}, eissn = {1306-6048}, address = {Prof. Dr. Abdullah HARMANCI}, year = {2014}, pages = {218 - 248}, doi = {10.24330/ieja.266249}, title = {CONSTRUCTION OF MODULES WITH A PRESCRIBED DIRECT SUM DECOMPOSITION}, key = {cite}, author = {Herbera, Dolors} }
APA Herbera, D . (2014). CONSTRUCTION OF MODULES WITH A PRESCRIBED DIRECT SUM DECOMPOSITION. International Electronic Journal of Algebra, 15 (15), 218-248. DOI: 10.24330/ieja.266249
MLA Herbera, D . "CONSTRUCTION OF MODULES WITH A PRESCRIBED DIRECT SUM DECOMPOSITION". International Electronic Journal of Algebra 15 (2014): 218-248 <http://dergipark.gov.tr/ieja/article/266249>
Chicago Herbera, D . "CONSTRUCTION OF MODULES WITH A PRESCRIBED DIRECT SUM DECOMPOSITION". International Electronic Journal of Algebra 15 (2014): 218-248
RIS TY - JOUR T1 - CONSTRUCTION OF MODULES WITH A PRESCRIBED DIRECT SUM DECOMPOSITION AU - Dolors Herbera Y1 - 2014 PY - 2014 N1 - doi: 10.24330/ieja.266249 DO - 10.24330/ieja.266249 T2 - International Electronic Journal of Algebra JF - Journal JO - JOR SP - 218 EP - 248 VL - 15 IS - 15 SN - 1306-6048-1306-6048 M3 - doi: 10.24330/ieja.266249 UR - http://dx.doi.org/10.24330/ieja.266249 Y2 - 2019 ER -
EndNote %0 International Electronic Journal of Algebra CONSTRUCTION OF MODULES WITH A PRESCRIBED DIRECT SUM DECOMPOSITION %A Dolors Herbera %T CONSTRUCTION OF MODULES WITH A PRESCRIBED DIRECT SUM DECOMPOSITION %D 2014 %J International Electronic Journal of Algebra %P 1306-6048-1306-6048 %V 15 %N 15 %R doi: 10.24330/ieja.266249 %U 10.24330/ieja.266249
ISNAD Herbera, Dolors . "CONSTRUCTION OF MODULES WITH A PRESCRIBED DIRECT SUM DECOMPOSITION". International Electronic Journal of Algebra 15 / 15 (Haziran 2014): 218-248. http://dx.doi.org/10.24330/ieja.266249
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