UNIQUENESS OF DECOMPOSITION, FACTORISATIONS, G-GROUPS AND POLYNOMIALS

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.

Keywords:

Krull-Schmidt Theorem, indecomposable module, direct-sum decomposition endomorphism ring, uniserial module,

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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.

International Electronic Journal of Algebra-Cover

ISSN: 1306-6048
Yayın Aralığı: Yılda 2 Sayı
Başlangıç: 2007
Yayıncı: Abdullah HARMANCI

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