Notes on quasi-Frobenius rings
-
Zhanmin Zhu
Abstract
We give some new characterizations of quasi-Frobenius rings. Namely, we prove that for a ring R, the following statements are equivalent:
(1) R is a quasi-Frobenius ring,
(2)
Funding source: Natural Science Foundation of Zhejiang Province
Award Identifier / Grant number: LY18A010018
Funding statement: This research was supported by the Natural Science Foundation of Zhejiang Province, China (LY18A010018).
References
[1] J. L. Chen and N. Q. Ding, On general principally injective rings, Comm. Algebra 27 (1999), no. 5, 2097–2116. 10.1080/00927879908826552Search in Google Scholar
[2]
J. L. Chen, N. Q. Ding, Y. L. Li and Y. Q. Zhou,
On
[3] J. L. Chen, Y. Q. Zhou and Z. M. Zhu, GP-injective rings need not be P-injective, Comm. Algebra 33 (2005), no. 7, 2395–2402. 10.1081/AGB-200058375Search in Google Scholar
[4] C. Faith and P. Menal, The structure of Johns rings, Proc. Amer. Math. Soc. 120 (1994), no. 4, 1071–1081. 10.1090/S0002-9939-1994-1231294-8Search in Google Scholar
[5] S. K. Jain, S. R. López-Permouth and S. T. Rizvi, Continuous rings with ACC on essentials are Artinian, Proc. Amer. Math. Soc. 108 (1990), no. 3, 583–586. 10.1090/S0002-9939-1990-0993754-0Search in Google Scholar
[6] S. B. Nam, N. K. Kim and J. Y. Kim, On simple GP-injective modules, Comm. Algebra 23 (1995), no. 14, 5437–5444. 10.1080/00927879508825543Search in Google Scholar
[7] W. K. Nicholson and M. F. Yousif, Principally injective rings, J. Algebra 174 (1995), no. 1, 77–93. 10.1017/CBO9780511546525.006Search in Google Scholar
[8] W. K. Nicholson and M. F. Yousif, Mininjective rings, J. Algebra 187 (1997), no. 2, 548–578. 10.1017/CBO9780511546525.003Search in Google Scholar
[9] W. K. Nicholson and M. F. Yousif, Annihilators and the CS-condition, Glasgow Math. J. 40 (1998), no. 2, 213–222. 10.1017/S0017089500032535Search in Google Scholar
[10] W. K. Nicholson and M. F. Yousif, Quasi-Frobenius Rings, Cambridge Tracts in Math. 158, Cambridge University, Cambridge, 2003. 10.1017/CBO9780511546525Search in Google Scholar
[11]
E. A. Rutter,
A characterization of
[12] E. A. Rutter, Jr., Rings with the principal extension property, Comm. Algebra 3 (1975), 203–212. 10.1080/00927877508822043Search in Google Scholar
[13] L. Shen and J. L. Chen, New characterizations of quasi-Frobenius rings, Comm. Algebra 34 (2006), no. 6, 2157–2165. 10.1080/00927870600549667Search in Google Scholar
[14] B. Stenström, Coherent rings and FP-injective modules, J. Lond. Math. Soc. (2) 2 (1970), 323–329. 10.1112/jlms/s2-2.2.323Search in Google Scholar
[15] R. Yue Chi Ming, On regular rings and self-injective rings. II, Glas. Mat. Ser. III 18(38) (1983), no. 2, 221–229. Search in Google Scholar
[16] R. Yue Chi Ming, On YJ-injectivity and annihilators, Georgian Math. J. 12 (2005), no. 3, 573–581. Search in Google Scholar
[17] Z. M. Zhu and J. L. Chen, 2-simple injective rings, Int. J. Algebra 4 (2010), no. 1–4, 25–37. Search in Google Scholar
[18] Z. M. Zhu, Some remarks on quasi-Frobenius rings, Georgian Math. J. 23 (2016), no. 1, 139–142. 10.1515/gmj-2015-0019Search in Google Scholar
© 2021 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Iterative rational least squares fitting
- Extensions of hom-Lie color algebras
- On the solvability of the modified Cauchy problem for linear systems of generalized ordinary differential equations with singularities
- On the solvability of a nonlocal problem for the system of Sobolev-type differential equations with integral condition
- Characterizing realcompact locales via remainders
- Jakimovski–Leviatan operators of Kantorovich type involving multiple Appell polynomials
- Further refinements of generalized numerical radius inequalities for Hilbert space operators
- An alternative transient solution for semi-Markov queuing systems
- On generalized fractional integration by parts formulas and their applications to boundary value problems
- Lie triple systems and Leibniz algebras
- Accessibility on iterated function systems
- Bell–Sheffer exponential polynomials of the second kind
- A study of differential prime rings with involution
- Existence result for nonlinear fractional differential equations with nonlocal fractional integro-differential boundary conditions in Banach spaces
- Inequalities for nonuniform wavelet frames
- Notes on quasi-Frobenius rings
Articles in the same Issue
- Frontmatter
- Iterative rational least squares fitting
- Extensions of hom-Lie color algebras
- On the solvability of the modified Cauchy problem for linear systems of generalized ordinary differential equations with singularities
- On the solvability of a nonlocal problem for the system of Sobolev-type differential equations with integral condition
- Characterizing realcompact locales via remainders
- Jakimovski–Leviatan operators of Kantorovich type involving multiple Appell polynomials
- Further refinements of generalized numerical radius inequalities for Hilbert space operators
- An alternative transient solution for semi-Markov queuing systems
- On generalized fractional integration by parts formulas and their applications to boundary value problems
- Lie triple systems and Leibniz algebras
- Accessibility on iterated function systems
- Bell–Sheffer exponential polynomials of the second kind
- A study of differential prime rings with involution
- Existence result for nonlinear fractional differential equations with nonlocal fractional integro-differential boundary conditions in Banach spaces
- Inequalities for nonuniform wavelet frames
- Notes on quasi-Frobenius rings
