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On some infinitely presented associative algebras

Published online by Cambridge University Press:  09 April 2009

Jacques Lewin
Jacques Lewin
Affiliation:
Syracuse UniversitySyracuse, NY, 13210, USA
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We prove here that if F is a finitely generated free associative algebra over the field and R is an ideal of F, then F/R2 is finitely presented if and only if F/R has finite dimension. Amitsur, [1, p. 136] asked whether a finitely generated algebra which is embeddable in matrices over a commutative f algebra is necessarily finitely presented. Let R = F′, the commutator ideal of F, then [4, theorem 6], F/F′2 is embeddable and thus provides a negative answer to his question. Another such example can be found in Small [6]. We also show that there are uncountably many two generator I algebras which satisfy a polynomial identity yet are not embeddable in any algebra of n xn matrices over a commutative algebra.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 16 , Issue 3 , November 1973 , pp. 290 - 293
Copyright
Copyright © Australian Mathematical Society 1973

References

[1]Amitsur, S. A., ‘A noncommutative Hilbert basis theorem and subrings of matrices’, Trans. Amer. Math. Soc. 149 (1970), 133–142.CrossRefGoogle Scholar
[2]Cohn, P. M, ‘On a generalization of the Euclidean algorithm’, Proc. Cambridge Phil. Soc., 57 (1961), 18–30.CrossRefGoogle Scholar
[3]Lewin, J., ‘Free modules over free algebras and free group algebras: The Schreier technique’, Trans. Amer. Math. Soc. 145, (1969) 455–465.CrossRefGoogle Scholar
[4]Lewin, J., ‘A matrix representation for associated algevras I.’, (to appear in Trans. Amer. Math. Soc.).Google Scholar
[5]Small, L., ‘An example in P. I. rings’, J. Algebra 17, (1971) 434–436.Google Scholar
[6]Small, L., ‘Ideals in finietly generated PI-Algebras’, Ring theory (Academic press) 1972, 347–352.Google Scholar