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Homogeneous elements of free algebras have free idealizers

Published online by Cambridge University Press:  24 October 2008

Warren Dicks
Warren Dicks
Affiliation:
Bedford College, London
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Abstract

Let k be a field, X a set, F = k 〈X〉 the free associative k-algebra, and b an element of F that is homogeneous with respect to the grading of F induced by some map . We show that the idealizer of b in F, S = {f∈F|fb∈bF}, is a free algebra.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 97 , Issue 1 , January 1985 , pp. 7 - 26
Copyright
Copyright © Cambridge Philosophical Society 1985

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References

REFERENCES

[1]Cohn, P. M.. Free Rings and their Relations. London Math. Soc. Monographs, no. 2 (Academic Press, 1971).Google Scholar
[2]Dicks, W.. On one-relator associative algebras. J. London Math. Soc. (2) 5 (1972), 249–252.CrossRefGoogle Scholar
[3]Dicks, W.. Idealizers in free algebras. Ph.D. Thesis, London, 1974.Google Scholar
[4]Dicks, W.. A free algebra can be free as a module over a non-free subalgebra. Bull. London Math. Soc. 15 (1982), 373–377.CrossRefGoogle Scholar
[5]Dicks, W.. On the cohomology of one-relator associative algebras. J. Algebra (in the Press).Google Scholar
[6]Gerasimov, V. N.. Distributive lattices of subspaces and the equality problem for algebras with a single relation. Algebra i Logika 15 (1976), 384–435 [Russian]. (English translation: Algebra and Logic 15 (1976), 238–274.)Google Scholar
[7]Lewin, J. and Lewin, T.. On ideals of free associative algebras generated by a single element. J. Algebra 8 (1968), 248–255.CrossRefGoogle Scholar
[8]Makar-Limanov, L.. On algebraically closed skew fields. J. Algebra (in the Press).Google Scholar