Three sets of conditions on rings
Author:
Abraham A. Klein
Journal:
Proc. Amer. Math. Soc. 25 (1970), 393-398
MSC:
Primary 16.50
DOI:
https://doi.org/10.1090/S0002-9939-1970-0263869-0
MathSciNet review:
0263869
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: We define a set of conditions ${\mathfrak {L}_m}$ on a ring $R$ using the notion of $R$-dependence of elements. We prove that ${\mathfrak {L}_1},{\mathfrak {L}_2}, \cdots$ is a strictly decreasing sequence of conditions. Two other sequences of conditions are considered and we prove that they are also strictly decreasing and we obtain their relation to ${\mathfrak {L}_m}$.
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G. M. Bergman, Commuting elements in free algebras, Doctoral Dissertation, Harvard University, Cambridge, Mass., 1968.
- P. M. Cohn, Free ideal rings, J. Algebra 1 (1964), 47–69. MR 161891, DOI https://doi.org/10.1016/0021-8693%2864%2990007-9
- Raouf Doss, Sur l’immersion d’un semi-groupe dans un groupe, Bull. Sci. Math. (2) 72 (1948), 139–150 (French). MR 29384
- Abraham A. Klein, Rings nonembeddable in fields with multiplicative semi-groups embeddable in groups, J. Algebra 7 (1967), 100–125. MR 230749, DOI https://doi.org/10.1016/0021-8693%2867%2990070-1
- Abraham A. Klein, Necessary conditions for embedding rings into fields, Trans. Amer. Math. Soc. 137 (1969), 141–151. MR 236212, DOI https://doi.org/10.1090/S0002-9947-1969-0236212-7
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Additional Information
Keywords:
Left and right <IMG WIDTH="21" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img3.gif" ALT="$R$">-dependent,
<IMG WIDTH="23" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$m$">-fir,
<IMG WIDTH="23" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img2.gif" ALT="$m$">-term algorithm,
formal series,
special series
Article copyright:
© Copyright 1970
American Mathematical Society