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Invariant ideals and polynomial forms. (English) Zbl 0998.16017
Let $$G$$ be a quasi-simple group of Lie type defined over an infinite field $$F$$ of characterstic $$p>0$$ and let $$V$$ be a finite dimensional vector space over a field $$E$$ of the same characteristic $$p$$. Assume that $$G$$ acts nontrivially on $$V$$ by way of the representation $$\phi\colon G\to\text{GL}(V)$$, and that $$V$$ contains no proper $$G$$-stable subgroup. If $$K$$ is a field of characteristic different from $$p$$, then the main result of this paper asserts that the augmentation ideal $$\omega K(V)$$ of the group algebra $$K[V]$$ is the unique proper $$G$$-stable ideal of $$K[V]$$. Thus the author completes the work of D. S. Passman and A. E. Zalesskij [Trans. Am. Math. Soc. 353, No. 7, 2971-2982 (2001; Zbl 0981.16024)] by studying the nonrational representations of $$G$$.

##### MSC:
 16S34 Group rings 20F50 Periodic groups; locally finite groups 20C07 Group rings of infinite groups and their modules (group-theoretic aspects) 16D25 Ideals in associative algebras 20G05 Representation theory for linear algebraic groups 20E32 Simple groups
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