Modules with semi-local endomorphism ring

Herbera, Dolors; Shamsuddin, Ahmad

doi:10.1090/S0002-9939-1995-1277114-8

Modules with semi-local endomorphism ring
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by Dolors Herbera and Ahmad Shamsuddin

Proc. Amer. Math. Soc. 123 (1995), 3593-3600

DOI: https://doi.org/10.1090/S0002-9939-1995-1277114-8

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Abstract:

We use the concept of dual Goldie dimension and a characterization of semi-local rings due to Camps and Dicks (1993) to find some classes of modules with semi-local endomorphism ring. We deduce that linearly compact modules have semi-local endomorphism ring, cancel from direct sums and satisfy the n th root uniqueness property. We also deduce that modules over commutative rings satisfying $AB{5^ \ast }$ also cancel from direct sums and satisfy the n th root uniqueness property.

References

Frank W. Anderson and Kent R. Fuller, Rings and categories of modules, 2nd ed., Graduate Texts in Mathematics, vol. 13, Springer-Verlag, New York, 1992. MR 1245487, DOI 10.1007/978-1-4612-4418-9
H. Bass, $K$-theory and stable algebra, Inst. Hautes Études Sci. Publ. Math. 22 (1964), 5–60. MR 174604
Rosa Camps and Warren Dicks, On semilocal rings, Israel J. Math. 81 (1993), no. 1-2, 203–211. MR 1231187, DOI 10.1007/BF02761306
Rosa Camps and Pere Menal, Power cancellation for Artinian modules, Comm. Algebra 19 (1991), no. 7, 2081–2095. MR 1121122, DOI 10.1080/00927879108824246
Friedrich Dischinger, Sur les anneaux fortement $\pi$-réguliers, C. R. Acad. Sci. Paris Sér. A-B 283 (1976), no. 8, Aii, A571–A573 (French, with English summary). MR 422330
E. Graham Evans Jr., Krull-Schmidt and cancellation over local rings, Pacific J. Math. 46 (1973), 115–121. MR 323815
Alberto Facchini, Dolors Herbera, Lawrence S. Levy, and Peter Vámos, Krull-Schmidt fails for Artinian modules, Proc. Amer. Math. Soc. 123 (1995), no. 12, 3587–3592. MR 1277109, DOI 10.1090/S0002-9939-1995-1277109-4
L. Fuchs, Torsion preradicals and ascending Loewy series of modules, J. Reine Angew. Math. 239(240) (1969), 169–179. MR 285565, DOI 10.1515/crll.1969.239-240.169
K. R. Goodearl, Surjective endomorphisms of finitely generated modules, Comm. Algebra 15 (1987), no. 3, 589–609. MR 882800, DOI 10.1080/00927878708823433
Azmi Hanna and Ahmad Shamsuddin, Duality in the category of modules, Algebra Berichte [Algebra Reports], vol. 49, Verlag Reinhard Fischer, Munich, 1984. Applications. MR 772927
C. R. Hajarnavis and N. C. Norton, On dual rings and their modules, J. Algebra 93 (1985), no. 2, 253–266. MR 786753, DOI 10.1016/0021-8693(85)90159-0
Christian U. Jensen and Helmut Lenzing, Model-theoretic algebra with particular emphasis on fields, rings, modules, Algebra, Logic and Applications, vol. 2, Gordon and Breach Science Publishers, New York, 1989. MR 1057608
Benoît Lemonnier, $\textrm {AB}5^{\ast }$ et la dualité de Morita, C. R. Acad. Sci. Paris Sér. A-B 289 (1979), no. 2, A47–A50 (French, with English summary). MR 549063
Horst Leptin, Linear kompakte Moduln und Ringe, Math. Z. 62 (1955), 241–267 (German). MR 69811, DOI 10.1007/BF01180634
Claudia Menini, Jacobson’s conjecture, Morita duality and related questions, J. Algebra 103 (1986), no. 2, 638–655. MR 864434, DOI 10.1016/0021-8693(86)90157-2
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
Bruno J. Müller, On Morita duality, Canadian J. Math. 21 (1969), 1338–1347. MR 255597, DOI 10.4153/CJM-1969-147-7
Bruno J. Müller, Linear compactness and Morita duality, J. Algebra 16 (1970), 60–66. MR 263875, DOI 10.1016/0021-8693(70)90040-2
F. L. Sandomierski, Linearly compact modules and local Morita duality, Ring theory (Proc. Conf., Park City, Utah, 1971) Academic Press, New York, 1972, pp. 333–346. MR 0344288
K. Varadarajan, Dual Goldie dimension, Comm. Algebra 7 (1979), no. 6, 565–610. MR 524269, DOI 10.1080/00927877908822364
Roger Ware, Endomorphism rings of projective modules, Trans. Amer. Math. Soc. 155 (1971), 233–256. MR 274511, DOI 10.1090/S0002-9947-1971-0274511-2
Weimin Xue, Rings with Morita duality, Lecture Notes in Mathematics, vol. 1523, Springer-Verlag, Berlin, 1992. MR 1184837, DOI 10.1007/BFb0089071
Daniel Zelinsky, Linearly compact modules and rings, Amer. J. Math. 75 (1953), 79–90. MR 51832, DOI 10.2307/2372616