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A Note on Annihilator Relations

Published online by Cambridge University Press:  22 January 2016

Yukitoshi Hinohara
Yukitoshi Hinohara*
Affiliation:
Tokyo Metropolitan University
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In a Frobenius algebra A over a field K, there exists a linear function λ of A into K which does not map any proper ideal of A onto 0. Then the map φ : x → xφ where

λ(xy) = λ(yxφ) for all y ε A,

defines an automorphism φ of A onto itself. This automorphism is called Nakayama’s automorphism. Now the following result is well known.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 17 , August 1960 , pp. 159 - 160
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1960

References

1) T. Nakayama, On Frobeniusean algebras II, Ann. of Math., 42 (1941), pp. 1-21.

2) This formulation of theorem is due to T. Nakayama. The writer’s original theorem was more special.

3) This theorem is valid if A is a semi-group with zero.

4) It is well known that this relation holds in a quasi-Frobenius rin.