Hostname: page-component-848d4c4894-nr4z6 Total loading time: 0 Render date: 2024-05-26T18:56:30.157Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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