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A NOTE ON ℵ₀-INJECTIVE RINGS

LIANG SHEN

Department of Mathematics, Southeast University, Nanjing 210096, P. R. China

https://doi.org/10.1142/S0219498811004677Cited by:2 (Source: Crossref)

Abstract

A ring R is called right ℵ₀-injective if every homomorphism from a countably generated right ideal of R to R_R can be extended to a homomorphism from R_R to R_R. In this note, some characterizations of ℵ₀-injective rings are given. It is proved that if R is semilocal, then R is right ℵ₀-injective if and only if every homomorphism from a countably generated small right ideal of R to R_R can be extended to one from R_R to R_R. It is also shown that if R is right noetherian and left ℵ₀-injective, then R is quasi-Frobenius. This result can be considered as an approach to the Faith–Menal conjecture.

Keywords:

AMSC: 16D50, 16L60

References

C. Faith and P. Menal, Proc. Amer. Math. Soc. 116, 21 (1992), DOI: 10.1090/S0002-9939-1992-1100651-0. Crossref, Web of Science, Google Scholar
C. Faith and P. Menal, Proc. Amer. Math. Soc. 120, 1071 (1994), DOI: 10.1090/S0002-9939-1994-1231294-8. Crossref, Web of Science, Google Scholar
J. L. Gómez Pardo and P. A. Guil Asensio, Abelian Groups Module Theory, and Topology, Lecture Notes in Pure and Applied Mathematics 201 (Marcel Dekker, New York, 1998) pp. 261–278. Google Scholar
B. Johns, J. Algebra 49, 222 (1977). Crossref, Web of Science, Google Scholar
K. Koike, Math. J. Okayama Univ. 37, 99 (1995). Web of Science, Google Scholar
R. P. Kurshan, J. Algebra 15, 376 (1970). Crossref, Web of Science, Google Scholar
C. Lanski, Canadian J. Math. 21, 904 (1969), DOI: 10.4153/CJM-1969-098-x. Crossref, Web of Science, Google Scholar
W. K. Nicholson, J. K. Park and M. F. Yousif, Commun. Algebra 28, 4665 (2000), DOI: 10.1080/00927870008827111. Crossref, Web of Science, Google Scholar
W. K. Nicholson and M. F. Yousif, J. Algebra 174, 77 (1995), DOI: 10.1006/jabr.1995.1117. Crossref, Web of Science, Google Scholar
W. K. Nicholson and M. F. Yousif , Quasi-Frobenius Rings ( Cambridge University Press , 2003 ) . Crossref, Google Scholar
E. A. Rutter, Commun. Algebra 3(3), 203 (1975), DOI: 10.1080/00927877508822043. Crossref, Web of Science, Google Scholar
L. Shen and J. L. Chen, Commun. Algebra 34, 2157 (2006), DOI: 10.1080/00927870600549667. Crossref, Web of Science, Google Scholar
A. A. Tuganbaev , Semidistributive Modules and Rings ( Kluwer Academic Publishers , 1998 ) . Crossref, Google Scholar