Abstract
In the early 1980s, Harada introduced extending and lifting properties for modules and, simultaneously, considered two new classes of artinian rings which contain QF-rings and Nakayama rings. The main purpose of this note is to survey his work and discuss its development and influence on the theory of rings and modules.
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References
K. Amdal and F. Ringdal, Categories uniserieles, C. R. Acad. Sci. Paris, Serie A 267 (1968), 85–86 and 247–249.
G. Azumaya, A duality theory for injective modules, Amer. J. Math. 81 (1959), 249–278.
Y. Baba, Injectivity of quasi-projective modules, projectivity of quasiinjective modules, and projective covers of injective modules, J. Algebra 155 (1993), 415–434.
Y. Baba and K. Oshiro, On a theorem of Fuller, J. Algebra 154 (1993), 86–94.
Y. Baba and K. Iwase, On quasi-Harada rings, J. Algebra 185 (1996), 544–570.
H. Bass Finitistic dimension and a homological generarization of semiprimary rings, Trans. Amer. Math. Soc. 95 (1960), 466–488.
J. Clark and R. Wisbauer, Σ-extending modules, J. Pure and Applied Algebra 104 (1995), 19–32.
Phan Dan, Right perfect rings with the extending property on finitely generated free modules, Osaka J. Math. 26 (1989), 265–273.
N. V. Dung, D. V. Huynh, P. F. Smith and R. Wisbauer, Extending Modules, Pitman, Harlow, 1994.
F. Dischinger and W. Müller, Einreihig zerlegbare artinsche Ringe sind selbstdual, Arch. Math. 48 (1984), 132–136.
C. Faith, Algebra II: Ring Theory, Springer-Verlag, Heidelberg-New York, 1976.
K. R. Fuller, On indecomposable injectives over artinian rings, Pacific J. Math. 29 (1969), 115–135.
K. R. Goodearl, Ring Theory, Nonsingular Rings and Modules,Pure Appl. Math. Series 33 Marcel Dekker, New York, 1976.
J. K. Haack, Self-duality and serial rings, J. Algebra 59 (1979), 345–363.
M. Harada, Nonsmall modules and noncosmall modules, Ring Theory, Proc. 1987 Antwerp Conf., Marcel Dekker, (1979), 669–690.
M. Harada, On one sided QF-2 rings. I, Osaka J. Math. 17 (1980), 421–431.
M. Harada, On one sided QF-2 rings. II, ibid., 433–438.
M. Harada Factor categories with applications to direct decomposition of modules, Lecture Notes Pure Appl. Math. 88 Marcel Dekker, New York, 1983.
M. Harada, On almost relative injectives of finite length, Preprint.
M. Harada and H. Kanbara, On categories of projective modules, Osaka J. Math. 8 (1971), 471–483.
M. Harada and T. Ishii, On perfect rings and the exchange property, Osaka J. Math. 12 (1975), 483–491.
M. Harada and K. Oshiro, On extending property on direct sums of uniform modules, Osaka J. Math. 18 (1981), 767–785.
M. Hoshino and T. Sumioka, Injective pairs in perfect rings, Osaka J. Math. 35 (1998), 501–508.
M. Hoshino, On strongly QF-rings, Communications in Algebra 28 (2000), 3585–3600.
L. Jeremy, Sur les modules et anneaux quasi-continus, Canad. Math. Bull. 17 (1974), 217–228.
D. V. Huynh and Phan Dan, Some characterizations of right co-Hrings,Math. J. Okayama Univ. 34 (1992), 165–174.
J. KadoThe maximal quotient rings of left H-rings, Osaka J. Math. 27 (1990), 247–252.
J. Kado and K. Oshiro, Self-duality and Harada rings, J. Algebra 211 (1999), 384–408.
T. Koike, An example of a QF-ring without a Nakayama automorphism and an example of a left H-ring without self-duality, in: “Proceedings of the 33rd Ring Theory and Representation Theory” vol. 33, Japan, 1999.
J. Kraemer, Characterizations of the existence of (quasi-) selfduality for complete tensor rings, Algebra-Berichte 56 Munchen, 1987.
H. Kupisch, Űber ein Klasse von Artin Ringen, Arch. Math. 26 (1975), 23–35.
T. Mano, The invariant system of serial rings and their applications to the theory of self-duality, Proc. 16th Symp. Ring Theory, Tokyo (1983), 48–53.
S. H. Mohamed and B. J. MĂĽller, Continuous modules and discrete modules,London Math. Soc. Lecture Notes 147 Cambridge Univ. Press, Cambridge, 1990.
K. Morita, Duality for modules and its applications to the theory of rings with minimal condition,Sci. Rep. Tokyo Kyoiku Daigaku Sect. A 6 (1958), 83–142.
M. Moromoto and T. Sumioka, Generalizations of theorems of Fuller, Osaka J. Math. 34 (1997), 689–701.
T. Nakayama, Note on uni-serial and generalized uni-serial rings,Proc. Imp. Acad. Tokyo 16 (1940), 285–289.
B. J. Müller and S. T. Rizvi, Direct sums of indecomposable modules, Osaka J. Math. 21 (1984), 365–374.
K. Oshiro, Semiperfect modules and quasi-semiperfect modules, Osaka J. Math. 20 (1983), 337–372.
K. Oshiro, Continuous modules and quasi-continuous modules, Osaka J. Math. 20 (1983), 681–694.
K. Oshiro, Lifting modules, extending modules and their applications to QF-rings, Hokkaido Math. J. 13 (1984), 310–338.
K. Oshiro, Lifting modules, extending modules and their applications to generalized uniserial rings, Hokkaido Math. J. 13 (1984), 339–346.
K. Oshiro, Structure of Nakayama rings, Proc. 20th Symp Ring Theory, Okayama, (1987), 109–133.
K. Oshiro, On Harada rings I, II, III,Math. J. Okayama Univ. 31 (1989), 161–178, 179–188; 32 (1990), 111–118.
K. Oshiro and K. Shigenaga, On H-rings with homogeneous socles, Math. J. Okayama Univ. 31 (1989), 189–196.
K. Oshiro and S. H. Rim, QF-rings with cyclic Nakayama permutation,Osaka J. Math. 34 (1997), 1–19.
S. T. Rizvi, Contributions to the theory of continuous modules, Ph.D. thesis, McMaster University, 1980.
T. Sumioka and S. Tozaki, On almost QF-rings, Osaka J. Math. 33 (1986), 649–661.
Y. Utumi, On continuous regular rings, Canad. Math. Bull. 4 (1961), 63–69.
N. Vanaja and V. M. Purav, Characterizations of generalized uniserial rings in terms of factor rings,Comm. Algebra 20 (1992), 2253–2270.
N. Vanaja, On Harada modules I, II, Preprint.
J. Waschbusch, Self-duality of serial rings, Comm Algebra 14 (1986), 581–590.
W. Xue, On a theorem of Fuller, J. Pure and Applied Algebra 122 (1997), 159–168.
W. Xue, Characterization of Morita duality via idempotents for semiperfect rings, Algebra Colloq. 5 (1998), 99–110.
K. Yamagata, Nakayama automorphisms and extension rings over algebras,in Japanese.
Y. Yukimoto, On decomposition of strongly quasi-Frobenius rings, Communications in Algebra 28 (2000), 1111–1114.
B. Zimmerman-Huisgen and W Zimmerman, Classes of modules with the exchange property,J. Algebra 88 (1984), 416–434.
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Oshiro, K. (2001). Theories of Harada in Artinian Rings and Applications to Classical Artinian Rings. In: Birkenmeier, G.F., Park, J.K., Park, Y.S. (eds) International Symposium on Ring Theory. Trends in Mathematics. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0181-6_21
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