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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Strong incompactness for some nonperfect rings


Author: Jan Trlifaj
Journal: Proc. Amer. Math. Soc. 123 (1995), 21-25
MSC: Primary 16D40; Secondary 03E75, 16E50
DOI: https://doi.org/10.1090/S0002-9939-1995-1212288-6
MathSciNet review: 1212288
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Abstract: Answering a question of Eklof and Mekler, for each regular uncountable cardinal $\kappa$, we construct a non-left-perfect ring ${R_\kappa }$ and a non-projective strongly $\kappa$-free left ideal ${I_\kappa }$ such that ${\text {gen}}({I_\kappa }) = \kappa$. Moreover, if $\kappa > {\aleph _1}$, then ${I_\kappa }$ is not $\kappa$-free. As consequences, we obtain results concerning incompactness spectra of non-perfect rings.


References [Enhancements On Off] (What's this?)

  • Frank W. Anderson and Kent R. Fuller, Rings and categories of modules, Springer-Verlag, New York-Heidelberg, 1974. Graduate Texts in Mathematics, Vol. 13. MR 0417223
  • Paul C. Eklof and Alan H. Mekler, Almost free modules, North-Holland Mathematical Library, vol. 46, North-Holland Publishing Co., Amsterdam, 1990. Set-theoretic methods. MR 1055083
  • K. R. Goodearl, von Neumann regular rings, Monographs and Studies in Mathematics, vol. 4, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1979. MR 533669

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Additional Information

Keywords: Almost free module, non-left-perfect ring, <IMG WIDTH="17" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$\kappa$">-free, strongly <IMG WIDTH="17" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img10.gif" ALT="$\kappa$">-free, incompactness spectrum
Article copyright: © Copyright 1995 American Mathematical Society