Exactness of the double dual
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- by R. R. Colby and K. R. Fuller PDF
- Proc. Amer. Math. Soc. 82 (1981), 521-526 Request permission
Abstract:
Let $R$ be an artinian ring. The double $R$-dual ( )** preserves monomorphisms, is left exact, or is right exact if and only if $R$ is QF-$3$, has dominant dimension at least 2, or is QF, respectively. Furthermore, both double dual functors preserve epimorphisms of finitely generated (left and right) modules if and only if both dual functors ( )* are exact on short exact sequences of finitely generated torsionless modules, and these conditions are equivalent to finitely generated torsionless modules being reflexive.References
- 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
- Stephen U. Chase, Direct products of modules, Trans. Amer. Math. Soc. 97 (1960), 457β473. MR 120260, DOI 10.1090/S0002-9947-1960-0120260-3
- R. R. Colby and E. A. Rutter Jr., Generalizations of $\textrm {QF}-3$ algebras, Trans. Amer. Math. Soc. 153 (1971), 371β386. MR 269686, DOI 10.1090/S0002-9947-1971-0269686-5
- Carl Faith, Rings with ascending condition on annihilators, Nagoya Math. J. 27 (1966), 179β191. MR 193107
- Kent R. Fuller, Generalized uniserial rings and their Kupisch series, Math. Z. 106 (1968), 248β260. MR 232795, DOI 10.1007/BF01110273
- Kent R. Fuller, On indecomposable injectives over artinian rings, Pacific J. Math. 29 (1969), 115β135. MR 246917
- Kiiti Morita, Duality in $\textrm {QF}-3$ rings, Math. Z. 108 (1969), 237β252. MR 241470, DOI 10.1007/BF01112025
- Bruno J. MΓΌller, The classification of algebras by dominant dimension, Canadian J. Math. 20 (1968), 398β409. MR 224656, DOI 10.4153/CJM-1968-037-9
- Edgar A. Rutter Jr., Two characterizations of quasi-Frobenius rings, Pacific J. Math. 30 (1969), 777β784. MR 248175
- Hiroyuki Tachikawa, Quasi-Frobenius rings and generalizations. $\textrm {QF}-3$ and $\textrm {QF}-1$ rings, Lecture Notes in Mathematics, Vol. 351, Springer-Verlag, Berlin-New York, 1973. Notes by Claus Michael Ringel. MR 0349740
- Hiroyuki Tachikawa, On algebras of which every indecomposable representation has an irreducible one as the top or the bottom Loewy constituent, Math. Z. 75 (1960/61), 215β227. MR 124356, DOI 10.1007/BF01211022
Additional Information
- © Copyright 1981 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 82 (1981), 521-526
- MSC: Primary 16A36
- DOI: https://doi.org/10.1090/S0002-9939-1981-0614871-5
- MathSciNet review: 614871