Every Σ-CS-module has an indecomposable decomposition

Gómez Pardo, José; Guil Asensio, Pedro

doi:10.1090/S0002-9939-00-05654-9

Every $\Sigma$-CS-module has an indecomposable decomposition
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by José L. Gómez Pardo and Pedro A. Guil Asensio PDF

Proc. Amer. Math. Soc. 129 (2001), 947-954 Request permission

Abstract:

We show that every $\Sigma$-CS module is a direct sum of uniform modules, thus solving an open problem posed in 1994 by Dung, Huynh, Smith and Wisbauer. With the help of this result we also answer several other questions related to indecomposable decompositions of CS-modules.

References

Ali Omer Al-attas and N. Vanaja, On $\Sigma$-extending modules, Comm. Algebra 25 (1997), no. 8, 2365–2393. MR 1459567, DOI 10.1080/00927879708825996
Frank W. Anderson and Kent R. Fuller, Rings and categories of modules, Graduate Texts in Mathematics, Vol. 13, Springer-Verlag, New York-Heidelberg, 1974. MR 0417223, DOI 10.1007/978-1-4684-9913-1
John Clark and Nguyen Viet Dung, On the decomposition of nonsingular CS-modules, Canad. Math. Bull. 39 (1996), no. 3, 257–265. MR 1411069, DOI 10.4153/CMB-1996-033-4
John Clark and Robert Wisbauer, $\Sigma$-extending modules, J. Pure Appl. Algebra 104 (1995), no. 1, 19–32. MR 1359688, DOI 10.1016/0022-4049(94)00119-4
—, Polyform and projective $\Sigma$-extending modules, Algebra Colloquium 5 (1998), 391–408.
Nguyen V. Dung, On indecomposable decompositions of CS-modules, J. Austral. Math. Soc. Ser. A 61 (1996), no. 1, 30–41. MR 1402111, DOI 10.1017/S1446788700000057
Nguyen Viet Dung, On indecomposable decompositions of CS-modules. II, J. Pure Appl. Algebra 119 (1997), no. 2, 139–153. MR 1453216, DOI 10.1016/S0022-4049(96)00056-4
Nguyen Viet Dung, Modules with indecomposable decompositions that complement maximal direct summands, J. Algebra 197 (1997), no. 2, 449–467. MR 1483773, DOI 10.1006/jabr.1997.7114
Nguyen Viet Dung, Dinh Van Huynh, Patrick F. Smith, and Robert Wisbauer, Extending modules, Pitman Research Notes in Mathematics Series, vol. 313, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1994. With the collaboration of John Clark and N. Vanaja. MR 1312366
Nguyen V. Dung and Patrick F. Smith, $\Sigma$-$\textrm {CS}$-modules, Comm. Algebra 22 (1994), no. 1, 83–93. MR 1255671, DOI 10.1080/00927879408824832
José L. Gómez Pardo and Pedro A. Guil Asensio, On the Goldie dimension of injective modules, Proc. Edinburgh Math. Soc. (2) 41 (1998), no. 2, 265–275. MR 1626488, DOI 10.1017/S0013091500019635
K. R. Goodearl, Singular torsion and the splitting properties, Memoirs of the American Mathematical Society, No. 124, American Mathematical Society, Providence, R.I., 1972. MR 0340335
Manabu Harada, Factor categories with applications to direct decomposition of modules, Lecture Notes in Pure and Applied Mathematics, vol. 88, Marcel Dekker, Inc., New York, 1983. MR 719882
Saad H. Mohamed and Bruno J. Müller, Continuous and discrete modules, London Mathematical Society Lecture Note Series, vol. 147, Cambridge University Press, Cambridge, 1990. MR 1084376, DOI 10.1017/CBO9780511600692
Morihiro Okado, On the decomposition of extending modules, Math. Japon. 29 (1984), no. 6, 939–941. MR 803451
Kiyoichi Oshiro, Lifting modules, extending modules and their applications to QF-rings, Hokkaido Math. J. 13 (1984), no. 3, 310–338. MR 764267, DOI 10.14492/hokmj/1381757705
B. L. Osofsky, Noninjective cyclic modules, Proc. Amer. Math. Soc. 19 (1968), 1383–1384. MR 231857, DOI 10.1090/S0002-9939-1968-0231857-7
Barbara L. Osofsky and Patrick F. Smith, Cyclic modules whose quotients have all complement submodules direct summands, J. Algebra 139 (1991), no. 2, 342–354. MR 1113780, DOI 10.1016/0021-8693(91)90298-M
R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach, Reading, 1991.