Balanced and 𝑄𝐹-1 algebras

Camillo, V. P.; Fuller, K. R.

doi:10.1090/S0002-9939-1972-0306256-0

Balanced and $QF-1$ algebras
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by V. P. Camillo and K. R. Fuller PDF

Proc. Amer. Math. Soc. 34 (1972), 373-378 Request permission

Abstract:

A ring R is QF-1 if every faithful module has the double centralizer property. It is proved that a local finite dimensional algebra is QF-1 if and only if it is QF. From this it follows that an arbitrary finite dimensional algebra has the property that every homomorphic image is QF-1 if and only if every homomorphic image is QF.

References

Victor P. Camillo, Balanced rings and a problem of Thrall, Trans. Amer. Math. Soc. 149 (1970), 143–153. MR 260794, DOI 10.1090/S0002-9947-1970-0260794-0
Charles W. Curtis and Irving Reiner, Representation theory of finite groups and associative algebras, Pure and Applied Mathematics, Vol. XI, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0144979
S. E. Dickson and K. R. Fuller, Commutative $\textrm {QF}-1$ artinian rings are $\textrm {QF}$, Proc. Amer. Math. Soc. 24 (1970), 667–670. MR 252426, DOI 10.1090/S0002-9939-1970-0252426-8
Carl Faith, A general Wedderburn theorem, Bull. Amer. Math. Soc. 73 (1967), 65–67. MR 202763, DOI 10.1090/S0002-9904-1967-11642-2
Denis Ragan Floyd, On $\textrm {QF}-1$ algebras, Pacific J. Math. 27 (1968), 81–94. MR 234988, DOI 10.2140/pjm.1968.27.81
Kent R. Fuller, Generalized uniserial rings and their Kupisch series, Math. Z. 106 (1968), 248–260. MR 232795, DOI 10.1007/BF01110273
Kent R. Fuller, Primary rings and double centralizers, Pacific J. Math. 34 (1970), 379–383. MR 271157, DOI 10.2140/pjm.1970.34.379
J. P. Jans, On the double centralizer condition, Math. Ann. 188 (1970), 85–89. MR 272817, DOI 10.1007/BF01350811
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
Kiiti Morita, On algebras for which every faithful representation is its own second commutator, Math. Z. 69 (1958), 429–434. MR 126492, DOI 10.1007/BF01187420
Kiiti Morita, On $S$-rings in the sense of F. Kasch, Nagoya Math. J. 27 (1966), 687–695. MR 199230, DOI 10.1017/S0027763000026477

rings

Tadasi Nakayama, On Frobeniusean algebras. II, Ann. of Math. (2) 42 (1941), 1–21. MR 4237, DOI 10.2307/1968984
Tadasi Nakayama, Note on uni-serial and generalized uni-serial rings, Proc. Imp. Acad. Tokyo 16 (1940), 285–289. MR 3618
C. J. Nesbitt and R. M. Thrall, Some ring theorems with applications to modular representations, Ann. of Math. (2) 47 (1946), 551–567. MR 16760, DOI 10.2307/1969092
R. M. Thrall, Some generalization of quasi-Frobenius algebras, Trans. Amer. Math. Soc. 64 (1948), 173–183. MR 26048, DOI 10.1090/S0002-9947-1948-0026048-0
Yasutaka Suzuki, A note on $\textrm {QF}-1$ algebras, Tohoku Math. J. (2) 22 (1970), 225–230. MR 269697, DOI 10.2748/tmj/1178242817