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DIRECT SUMS OF INFINITELY MANY KERNELS

  • ŞULE ECEVIT (a1), ALBERTO FACCHINI (a2) and M. TAMER KOŞAN (a3)

Abstract

Let 𝒦 be the class of all right R-modules that are kernels of nonzero homomorphisms φ:E1E2 for some pair of indecomposable injective right R-modules E1,E2. In a previous paper, we completely characterized when two direct sums A1⊕⋯⊕An and B1⊕⋯⊕Bm of finitely many modules Ai and Bj in 𝒦 are isomorphic. Here we consider the case in which there are arbitrarily, possibly infinitely, many Ai and Bj in 𝒦. In both the finite and the infinite case, the behaviour is very similar to that which occurs if we substitute the class 𝒦 with the class 𝒰 of all uniserial right R-modules (a module is uniserial when its lattice of submodules is linearly ordered).

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Copyright

Corresponding author

For correspondence; e-mail: facchini@math.unipd.it

Footnotes

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Alberto Facchini was partially supported by the Italian Ministero dell’Istruzione, dell’Università e della Ricerca (Prin 2007 ‘Rings, algebras, modules and categories’) and by the Università di Padova (Progetto di Ricerca di Ateneo CPDA071244/07).

Footnotes

References

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[1]Amini, B., Amini, A. and Facchini, A., ‘Equivalence of diagonal matrices over local rings’, J. Algebra 320 (2008), 12881310.
[2]Amini, B., Amini, A. and Facchini, A., ‘Direct summands of direct sums of modules whose endomorphism rings have two maximal right ideals’, manuscript (2010).
[3]Bumby, R. T., ‘Modules which are isomorphic to submodules of each other’, Arch. Math. 16 (1965), 184185.
[4]Dung, N. V. and Facchini, A., ‘Weak Krull–Schmidt for infinite direct sums of uniserial modules’, J. Algebra 193 (1997), 102121.
[5]Facchini, A., ‘Krull–Schmidt fails for serial modules’, Trans. Amer. Math. Soc. 348 (1996), 45614575.
[6]Facchini, A., Module Theory. Endomorphism Rings and Direct Sum Decompositions in Some Classes of Modules, Progress in Mathematics, 167 (Birkhäuser, Basel, 1998).
[7]Facchini, A., ‘Injective modules, spectral categories, and applications’, in: Noncommutative Rings, Group Rings, Diagram Algebras and Their Applications, Contemporary Mathematics, 456 (eds. Jain, S. K. and Parvathi, S.) (American Mathematical Society, Providence, RI, 2008), pp. 117.
[8]Facchini, A., Ecevit, Ş. and Tamer Koşan, M., ‘Kernels of morphisms between indecomposable injective modules’, Glasgow Math. J. (2010), to appear.
[9]Facchini, A. and Girardi, N., ‘Couniformly presented modules and dualities’, in: Advances in Ring Theory, Trends in Mathematics (eds. Huynh, Dinh Van and López Permouth, Sergio R.) (Birkhäuser, Basel, 2010), pp. 149163.
[10]Facchini, A. and Příhoda, P., ‘Representations of the category of serial modules of finite Goldie dimension’, in: Models, Modules and Abelian Groups (eds. Göbel, R. and Goldsmith, B.) (de Gruyter, Berlin, 2008), pp. 463486.
[11]Facchini, A. and Příhoda, P., ‘Factor categories and infinite direct sums’, Int. Electron. J. Algebra 5 (2009), 134.
[12]Gabriel, P. and Oberst, U., ‘Spektralkategorien und reguläre Ringe im Von-Neumannschen Sinn’, Math. Z. 82 (1966), 389395.
[13]Herzog, I., ‘Contravariant functors on the category of finitely presented modules’, Israel J. Math. 167 (2008), 347410.
[14]Příhoda, P., ‘A version of the weak Krull–Schmidt theorem for infinite direct sums of uniserial modules’, Comm. Algebra 34 (2006), 14791487.
[15]Puninski, G., ‘Some model theory over a nearly simple uniserial domain and decompositions of serial modules’, J. Pure Appl. Algebra 163 (2001), 319337.
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DIRECT SUMS OF INFINITELY MANY KERNELS

  • ŞULE ECEVIT (a1), ALBERTO FACCHINI (a2) and M. TAMER KOŞAN (a3)

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