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Notes on quasi-Frobenius rings

  • Zhanmin Zhu EMAIL logo
Published/Copyright: November 22, 2019
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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) M2(R) is right Johns and every closed left ideal of R is cyclic, (3) R is a left 2-simple injective left Kasch ring with ACC on left annihilators, (4) R is a left 2-injective semilocal ring such that R/Sl is left Goldie, (5) R is a right YJ-injective right minannihilator ring with ACC on right annihilators.

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 (m,n)-injectivity of modules, Comm. Algebra 29 (2001), no. 12, 5589–5603. 10.1081/AGB-100107948Search in Google Scholar

[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 QF-3 rings, Pacific J. Math. 51 (1974), 533–536. 10.2140/pjm.1974.51.533Search in Google Scholar

[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

Received: 2016-10-19
Accepted: 2019-05-15
Published Online: 2019-11-22
Published in Print: 2021-02-01

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