Research Article
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Year 2018, , 107 - 128, 05.07.2018
https://doi.org/10.24330/ieja.440235

Abstract

References

  • A. Alahmadi and A. Facchini, Direct products of modules whose endomorphism rings have at most two maximal ideals, J. Algebra, 435 (2015), 204-222.
  • B. Amini, A. Amini and A. Facchini, Equivalence of diagonal matrices over local rings, J. Algebra, 320(3) (2008), 1288-1310.
  • A. Amini, B. Amini and A. Facchini, Weak Krull-Schmidt for in nite direct sums of cyclically presented modules over local rings, Rend. Semin. Mat. Univ. Padova, 122 (2009), 39-54.
  • F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, New York, Springer-Verlag, 1974.
  • M. J. Arroyo Paniagua and A. Facchini, G-groups and biuniform abelian normal subgroups, Adv. Group Theory Appl., 2 (2016), 79-111.
  • G. Azumaya, Corrections and supplementaries to my paper concerning Krull- Remak-Schmidt's theorem, Nagoya Math. J., 1 (1950), 117-124.
  • G. M. Bergman, Coproducts and some universal ring constructions, Trans. Amer. Math. Soc., 200 (1974), 33-88.
  • G. M. Bergman and W. Dicks, Universal derivations and universal ring constructions, Pac. J. Math., 79 (1978), 293-337.
  • H.-H. Brungs, Ringe mit eindeutiger Faktorzerlegung, J. Reine Angew. Math., 236 (1969), 43-66.
  • F. Campanini, On a category of chains of modules whose endomorphism rings have at most 2n maximal right ideals, Comm. Algebra, 46(5) (2018), 1971-1982.
  • F. Campanini and A. Facchini, On a category of extensions whose endomorphism rings have at most four maximal ideals, to appear in \Advances in Rings and Modules, S. Lopez-Permouth, J. K. Park, C. Roman and S. T. Rizvi Eds, Contemp. Math., 2018.
  • J. Coykendall and W. W. Smith, On unique factorisation domains, J. Algebra, 332 (2011), 62-70.
  • N. V. Dung and A. Facchini, Weak Krull-Schmidt for in nite direct sums of uniserial modules, J. Algebra, 193 (1997), 102-121.
  • S. Ecevit, A. Facchini, and M. T. Kosan, Direct sums of in nitely many kernels, J. Aust. Math. Soc., 89(2) (2010), 199-214.
  • [A. Facchini, Krull-Schmidt fails for serial modules, Trans. Amer. Math. Soc., 348(11) (1996), 4561-4575.
  • A. Facchini, Module Theory. Endomorphism Rings and Direct Sum Decompositions in Some Classes of Modules, Birkhauser Verlag, Basel, 1998.
  • A. Facchini, Direct-sum decompositions of modules with semilocal endomorphism rings, Bull. Math. Sci., 2(2) (2012), 225-279.
  • A. Facchini and M. Altun-  Ozarslan, The Krull-Schmidt-Remak-Azumaya Theorem for G-groups, to appear in the proceedings of the Conference \Noncommutative rings and their applications, V", Lens 12-15 June 2017, Contemp. Math., 2018.
  • A. Facchini, S. Ecevit and M. T. Kosan, Kernels of morphisms between indecomposable injective modules, Glasg. Math. J., 52(A) (2010), 69-82.
  • A. Facchini and N. Girardi, Couniformly presented modules and dualities, Advances in ring theory, Trends Math., Birkh^auser/Springer Basel AG, Basel, (2010), 149-164.
  • A. Facchini and Z. Nazemian, Equivalence of some homological conditions for ring epimorphisms, to appear in J. Pure Appl. Algebra, 2018.
  • A. Facchini and P. Prhoda, The Krull-Schmidt theorem in the case two, Algebr. Represent. Theory, 14(3) (2011), 545-570.
  • H. Frobenius and H. Stickelberger,  Uber Gruppen von vertauschbaren Elementen, J. Reine Angew. Math., 86 (1879), 217-262.
  • J. Hashimoto, On direct product decomposition of partially ordered sets, Ann. of Math., 54(2) (1951), 315-318.
  • W. Krull,  Uber verallgemeinerte endliche Abelsche Gruppen, Math. Z., 23(1) (1925), 161-196.
  • E. L. Lady, Summands of nite rank torsion-free abelian groups, J. Algebra, 32 (1974), 51-52.
  • J. H. Maclagan-Wedderburn, On the direct product in the theory of nite groups, Ann. of Math., 10(4) (1909), 173-176.
  • T. Nakayama, and J. Hashimoto, On a problem of G. Birkho , Proc. Amer. Math. Soc., 1 (1950), 141-142.
  • P. Prhoda, A version of the weak Krull-Schmidt theorem for in nite direct sums of uniserial modules, Comm. Algebra, 34(4) (2006), 1479-1487.
  • P. Prhoda, Add(U) of a uniserial module, Comment. Math. Univ. Carolin., 47(3) (2006), 391-398.
  • G. Puninski, Some model theory over a nearly simple uniserial domain and decompositions of serial modules, J. Pure Appl. Algebra, 163(3) (2001), 319- 337.
  • R. E. Remak,  Uber die Zerlegung der endlichen Gruppen in indirekte unzerlegbare faktoren, dissertation, 1911.
  • O. Y. Schmidt, Sur les produits directs, Bull. Soc. Math. France, 41 (1913), 161-164.
  • O. Y. Schmidt,  Uber unendliche Gruppen mit endlicher Kette, Math. Z., 29 (1929), 34-41.
  • M. Suzuki, Group Theory I, Springer-Verlag, Berlin-New York, 1982. [36] R. B.War eld, Purity and algebraic compactness for modules, Paci c J. Math., 28 (1969), 699-719.
  • R. B. War eld, Serial rings and nitely presented modules, J. Algebra, 37 (1975), 187-222.

UNIQUENESS OF DECOMPOSITION, FACTORISATIONS, G-GROUPS AND POLYNOMIALS

Year 2018, , 107 - 128, 05.07.2018
https://doi.org/10.24330/ieja.440235

Abstract

In this article, we present the classical Krull-Schmidt Theorem for groups, its statement for modules due to Azumaya, and much more modern variations on the theme, like the so-called weak Krull-Schmidt Theorem, which holds for some particular classes of modules. Also, direct product of modules is considered. We present some properties of the category of G-groups, a category in which Remak's results about the Krull-Schmidt Theorem for groups can be better understood. In the last section, direct-sum decompositions and factorisations in other algebraic structures are considered.

References

  • A. Alahmadi and A. Facchini, Direct products of modules whose endomorphism rings have at most two maximal ideals, J. Algebra, 435 (2015), 204-222.
  • B. Amini, A. Amini and A. Facchini, Equivalence of diagonal matrices over local rings, J. Algebra, 320(3) (2008), 1288-1310.
  • A. Amini, B. Amini and A. Facchini, Weak Krull-Schmidt for in nite direct sums of cyclically presented modules over local rings, Rend. Semin. Mat. Univ. Padova, 122 (2009), 39-54.
  • F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, New York, Springer-Verlag, 1974.
  • M. J. Arroyo Paniagua and A. Facchini, G-groups and biuniform abelian normal subgroups, Adv. Group Theory Appl., 2 (2016), 79-111.
  • G. Azumaya, Corrections and supplementaries to my paper concerning Krull- Remak-Schmidt's theorem, Nagoya Math. J., 1 (1950), 117-124.
  • G. M. Bergman, Coproducts and some universal ring constructions, Trans. Amer. Math. Soc., 200 (1974), 33-88.
  • G. M. Bergman and W. Dicks, Universal derivations and universal ring constructions, Pac. J. Math., 79 (1978), 293-337.
  • H.-H. Brungs, Ringe mit eindeutiger Faktorzerlegung, J. Reine Angew. Math., 236 (1969), 43-66.
  • F. Campanini, On a category of chains of modules whose endomorphism rings have at most 2n maximal right ideals, Comm. Algebra, 46(5) (2018), 1971-1982.
  • F. Campanini and A. Facchini, On a category of extensions whose endomorphism rings have at most four maximal ideals, to appear in \Advances in Rings and Modules, S. Lopez-Permouth, J. K. Park, C. Roman and S. T. Rizvi Eds, Contemp. Math., 2018.
  • J. Coykendall and W. W. Smith, On unique factorisation domains, J. Algebra, 332 (2011), 62-70.
  • N. V. Dung and A. Facchini, Weak Krull-Schmidt for in nite direct sums of uniserial modules, J. Algebra, 193 (1997), 102-121.
  • S. Ecevit, A. Facchini, and M. T. Kosan, Direct sums of in nitely many kernels, J. Aust. Math. Soc., 89(2) (2010), 199-214.
  • [A. Facchini, Krull-Schmidt fails for serial modules, Trans. Amer. Math. Soc., 348(11) (1996), 4561-4575.
  • A. Facchini, Module Theory. Endomorphism Rings and Direct Sum Decompositions in Some Classes of Modules, Birkhauser Verlag, Basel, 1998.
  • A. Facchini, Direct-sum decompositions of modules with semilocal endomorphism rings, Bull. Math. Sci., 2(2) (2012), 225-279.
  • A. Facchini and M. Altun-  Ozarslan, The Krull-Schmidt-Remak-Azumaya Theorem for G-groups, to appear in the proceedings of the Conference \Noncommutative rings and their applications, V", Lens 12-15 June 2017, Contemp. Math., 2018.
  • A. Facchini, S. Ecevit and M. T. Kosan, Kernels of morphisms between indecomposable injective modules, Glasg. Math. J., 52(A) (2010), 69-82.
  • A. Facchini and N. Girardi, Couniformly presented modules and dualities, Advances in ring theory, Trends Math., Birkh^auser/Springer Basel AG, Basel, (2010), 149-164.
  • A. Facchini and Z. Nazemian, Equivalence of some homological conditions for ring epimorphisms, to appear in J. Pure Appl. Algebra, 2018.
  • A. Facchini and P. Prhoda, The Krull-Schmidt theorem in the case two, Algebr. Represent. Theory, 14(3) (2011), 545-570.
  • H. Frobenius and H. Stickelberger,  Uber Gruppen von vertauschbaren Elementen, J. Reine Angew. Math., 86 (1879), 217-262.
  • J. Hashimoto, On direct product decomposition of partially ordered sets, Ann. of Math., 54(2) (1951), 315-318.
  • W. Krull,  Uber verallgemeinerte endliche Abelsche Gruppen, Math. Z., 23(1) (1925), 161-196.
  • E. L. Lady, Summands of nite rank torsion-free abelian groups, J. Algebra, 32 (1974), 51-52.
  • J. H. Maclagan-Wedderburn, On the direct product in the theory of nite groups, Ann. of Math., 10(4) (1909), 173-176.
  • T. Nakayama, and J. Hashimoto, On a problem of G. Birkho , Proc. Amer. Math. Soc., 1 (1950), 141-142.
  • P. Prhoda, A version of the weak Krull-Schmidt theorem for in nite direct sums of uniserial modules, Comm. Algebra, 34(4) (2006), 1479-1487.
  • P. Prhoda, Add(U) of a uniserial module, Comment. Math. Univ. Carolin., 47(3) (2006), 391-398.
  • G. Puninski, Some model theory over a nearly simple uniserial domain and decompositions of serial modules, J. Pure Appl. Algebra, 163(3) (2001), 319- 337.
  • R. E. Remak,  Uber die Zerlegung der endlichen Gruppen in indirekte unzerlegbare faktoren, dissertation, 1911.
  • O. Y. Schmidt, Sur les produits directs, Bull. Soc. Math. France, 41 (1913), 161-164.
  • O. Y. Schmidt,  Uber unendliche Gruppen mit endlicher Kette, Math. Z., 29 (1929), 34-41.
  • M. Suzuki, Group Theory I, Springer-Verlag, Berlin-New York, 1982. [36] R. B.War eld, Purity and algebraic compactness for modules, Paci c J. Math., 28 (1969), 699-719.
  • R. B. War eld, Serial rings and nitely presented modules, J. Algebra, 37 (1975), 187-222.
There are 36 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Alberto Facchini

Serap Sahinkaya

Publication Date July 5, 2018
Published in Issue Year 2018

Cite

APA Facchini, A., & Sahinkaya, S. (2018). UNIQUENESS OF DECOMPOSITION, FACTORISATIONS, G-GROUPS AND POLYNOMIALS. International Electronic Journal of Algebra, 24(24), 107-128. https://doi.org/10.24330/ieja.440235
AMA Facchini A, Sahinkaya S. UNIQUENESS OF DECOMPOSITION, FACTORISATIONS, G-GROUPS AND POLYNOMIALS. IEJA. July 2018;24(24):107-128. doi:10.24330/ieja.440235
Chicago Facchini, Alberto, and Serap Sahinkaya. “UNIQUENESS OF DECOMPOSITION, FACTORISATIONS, G-GROUPS AND POLYNOMIALS”. International Electronic Journal of Algebra 24, no. 24 (July 2018): 107-28. https://doi.org/10.24330/ieja.440235.
EndNote Facchini A, Sahinkaya S (July 1, 2018) UNIQUENESS OF DECOMPOSITION, FACTORISATIONS, G-GROUPS AND POLYNOMIALS. International Electronic Journal of Algebra 24 24 107–128.
IEEE A. Facchini and S. Sahinkaya, “UNIQUENESS OF DECOMPOSITION, FACTORISATIONS, G-GROUPS AND POLYNOMIALS”, IEJA, vol. 24, no. 24, pp. 107–128, 2018, doi: 10.24330/ieja.440235.
ISNAD Facchini, Alberto - Sahinkaya, Serap. “UNIQUENESS OF DECOMPOSITION, FACTORISATIONS, G-GROUPS AND POLYNOMIALS”. International Electronic Journal of Algebra 24/24 (July 2018), 107-128. https://doi.org/10.24330/ieja.440235.
JAMA Facchini A, Sahinkaya S. UNIQUENESS OF DECOMPOSITION, FACTORISATIONS, G-GROUPS AND POLYNOMIALS. IEJA. 2018;24:107–128.
MLA Facchini, Alberto and Serap Sahinkaya. “UNIQUENESS OF DECOMPOSITION, FACTORISATIONS, G-GROUPS AND POLYNOMIALS”. International Electronic Journal of Algebra, vol. 24, no. 24, 2018, pp. 107-28, doi:10.24330/ieja.440235.
Vancouver Facchini A, Sahinkaya S. UNIQUENESS OF DECOMPOSITION, FACTORISATIONS, G-GROUPS AND POLYNOMIALS. IEJA. 2018;24(24):107-28.