Self-injective rings

Faith, Carl

doi:10.1090/S0002-9939-1979-0542077-8

Self-injective rings
HTML articles powered by AMS MathViewer

by Carl Faith

Proc. Amer. Math. Soc. 77 (1979), 157-164

DOI: https://doi.org/10.1090/S0002-9939-1979-0542077-8

PDF | Request permission

Abstract:

In 1958 Matlis proved that the study of Noetherian complete local rings could be subsumed under the study of injective modules E over a commutative ring A such that $B = \mathrm {End}_A E$ is commutative. In this case $B = \text {End}_B E$, and E is said to be strongly balanced over B. The main theorem of this paper shows that the study of strongly balanced injectives over any ring, and hence the study of Monta self-dualities, is contained in the study of self-injective rings.

References

Goro Azumaya, A duality theory for injective modules. (Theory of quasi-Frobenius modules), Amer. J. Math. 81 (1959), 249–278. MR 106932, DOI 10.2307/2372855
Goro Azumaya, Completely faithful modules and self-injective rings, Nagoya Math. J. 27 (1966), 697–708. MR 213389
Carl Faith, Algebra: rings, modules and categories. I, Die Grundlehren der mathematischen Wissenschaften, Band 190, Springer-Verlag, New York-Heidelberg, 1973. MR 0366960
Carl Faith, Algebra. II, Grundlehren der Mathematischen Wissenschaften, No. 191, Springer-Verlag, Berlin-New York, 1976. Ring theory. MR 0427349
Carl Faith, Injective cogenerator rings and a theorem of Tachikawa, Proc. Amer. Math. Soc. 60 (1976), 25–30 (1977). MR 417237, DOI 10.1090/S0002-9939-1976-0417237-4
Carl Faith, Injective cogenerator rings and a theorem of Tachikawa. II, Proc. Amer. Math. Soc. 62 (1976), no. 1, 15–18 (1977). MR 429990, DOI 10.1090/S0002-9939-1977-0429990-5
D. T. Gill, Almost maximal valuation rings, J. London Math. Soc. (2) 4 (1971), 140–146. MR 292822, DOI 10.1112/jlms/s2-4.1.140
Masatosi Ikeda and Tadasi Nakayama, On some characteristic properties of quasi-Frobenius and regular rings, Proc. Amer. Math. Soc. 5 (1954), 15–19. MR 60489, DOI 10.1090/S0002-9939-1954-0060489-9
Arun Vinayak Jategaonkar, Certain injectives are Artinian, Noncommutative ring theory (Internat. Conf., Kent State Univ., Kent, Ohio, 1975) Lecture Notes in Math., Vol. 545, Springer, Berlin, 1976, pp. 128–139. MR 0432703
Irving Kaplansky, Maximal fields with valuations, Duke Math. J. 9 (1942), 303–321. MR 6161
Irving Kaplansky, Modules over Dedekind rings and valuation rings, Trans. Amer. Math. Soc. 72 (1952), 327–340. MR 46349, DOI 10.1090/S0002-9947-1952-0046349-0
Toyonori Kato, Self-injective rings, Tohoku Math. J. (2) 19 (1967), 485–495. MR 224648, DOI 10.2748/tmj/1178243253
Lawrence S. Levy, Commutative rings whose homomorphic images are self-injective, Pacific J. Math. 18 (1966), 149–153. MR 194453
G. B. Klatt and L. S. Levy, Pre-self-injective rings, Trans. Amer. Math. Soc. 137 (1969), 407–419. MR 236165, DOI 10.1090/S0002-9947-1969-0236165-1
Eben Matlis, Injective modules over Noetherian rings, Pacific J. Math. 8 (1958), 511–528. MR 99360
Eben Matlis, Injective modules over Prüfer rings, Nagoya Math. J. 15 (1959), 57–69. MR 109840
Kiiti Morita, Duality for modules and its applications to the theory of rings with minimum condition, Sci. Rep. Tokyo Kyoiku Daigaku Sect. A 6 (1958), 83–142. MR 96700
Takesi Onodera, Über Kogeneratoren, Arch. Math. (Basel) 19 (1968), 402–410 (German). MR 233849, DOI 10.1007/BF01898421
B. L. Osofsky, A generalization of quasi-Frobenius rings, J. Algebra 4 (1966), 373–387. MR 204463, DOI 10.1016/0021-8693(66)90028-7
Bo Stenström, Coherent rings and $F\,P$-injective modules, J. London Math. Soc. (2) 2 (1970), 323–329. MR 258888, DOI 10.1112/jlms/s2-2.2.323
Peter Vámos, Classical rings, J. Algebra 34 (1975), 114–129. MR 382250, DOI 10.1016/0021-8693(75)90198-2
P. Vámos, Rings with duality, Proc. London Math. Soc. (3) 35 (1977), no. 2, 275–289. MR 450324, DOI 10.1112/plms/s3-35.2.275
P. Vámos, Semi-local Noetherian $PI$-rings, Bull. London Math. Soc. 9 (1977), no. 3, 251–256. MR 460382, DOI 10.1112/blms/9.3.251

FP-injective rings and modules

Bruno J. Müller, On Morita duality, Canadian J. Math. 21 (1969), 1338–1347. MR 255597, DOI 10.4153/CJM-1969-147-7

Localization, completion and duality in HNP rings

Saroj Jain, Flat and FP-injectivity, Proc. Amer. Math. Soc. 41 (1973), 437–442. MR 323828, DOI 10.1090/S0002-9939-1973-0323828-9
V. S. Ramamurthi and K. M. Rangaswamy, On finitely injective modules, J. Austral. Math. Soc. 16 (1973), 239–248. Collection of articles dedicated to the memory of Hanna Neumann, II. MR 0332882
Hideaki Sekiyama, Trivial extension of a ring with balanced condition, Proc. Amer. Math. Soc. 77 (1979), no. 1, 1–6. MR 539618, DOI 10.1090/S0002-9939-1979-0539618-3
Robert M. Fossum, Phillip A. Griffith, and Idun Reiten, Trivial extensions of abelian categories, Lecture Notes in Mathematics, Vol. 456, Springer-Verlag, Berlin-New York, 1975. Homological algebra of trivial extensions of abelian categories with applications to ring theory. MR 0389981