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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Invariants of Skew Derivations


Authors: Jeffrey Bergen and Piotr Grzeszczuk
Journal: Proc. Amer. Math. Soc. 125 (1997), 3481-3488
MSC (1991): Primary 16W20, 16W25, 16W55
MathSciNet review: 1415574
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Abstract: If $\sigma $ is an automorphism and $\delta $ is a $\sigma $-derivation of a ring $R$, then the subring of invariants is the set $R^{(\delta )} = \{r \in R \mid \delta (r) = 0 \}.$ The main result of this paper is

Theorem. Let $\delta $ be a $\sigma $-derivation of an algebra $R$ over a commutative ring $K$ such that

\begin{equation*}\delta ^{n+k}(r) + a_{n-1}\delta ^{n+k-1}(r) + \dots + a_{1}\delta ^{k+1}(r) + a_{0}\delta ^{k}(r) =0, \end{equation*}

for all $r \in R$, where $a_{n-1}, \dots , a_{1},a_{0} \in K$ and ${a_{0}}^{-1} \in K$.

(i)
If $R^{n+1} \not = 0$, then $R^{(\delta )} \not = 0$.
(ii)
If $L$ is a $\delta $-stable left ideal of $R$ such that $l.ann_{R}(L) = 0$, then $L^{(\delta )} \not = 0$.

This theorem generalizes results on the invariants of automorphisms and derivations.


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Additional Information

Jeffrey Bergen
Affiliation: Institute of Mathematics, University of Warsaw, Białystok Division Akademicka 2, 15-267, Białystok, Poland
Email: jbergen@condor.depaul.edu

Piotr Grzeszczuk
Affiliation: Institute of Mathematics, University of Warsaw, Białystok Division Akademicka 2, 15-267, Białystok, Poland
Email: piotrgr@cksr.ac.bialystok.pl

DOI: http://dx.doi.org/10.1090/S0002-9939-97-04045-8
PII: S 0002-9939(97)04045-8
Received by editor(s): December 29, 1995
Received by editor(s) in revised form: July 2, 1996
Additional Notes: The first author was supported by the University Research Council at DePaul University. Both authors were supported by Polish KBN Grant 2 PO3A 050 08. Much of this work was done when the first author was a visitor at the University of Warsaw, Białystok Division and the second author was a visitor at DePaul University. We would like to thank both universities for their hospitality
Communicated by: Ken Goodearl
Article copyright: © Copyright 1997 American Mathematical Society