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On the automorphisms of the group ring of a unique product group. (English) Zbl 0582.16002

Let R be a ring with an identity and a nilpotent ideal N. Let G be a group and let R(G) be the group ring of G over R. An automorphism of G extends naturally to an R-linear, augmentation-preserving, automorphism of R(G) but not every such automorphism of R(G) is so obtainable. We show that if R/N is a domain, if G is a unique product group (or, more generally, if R/N(G) has no non-trivial units) and if \(\theta\) is an automorphism of R(G) which is augmentation-preserving modulo N then there exists an automorphism \(\phi\) of G such that \(\theta\) (g)\(\equiv \phi (g) mod N(G)\) (\(\forall g\in G)\) and that, in a strong sense, a converse to this assertion holds.

MSC:

16S34 Group rings
16W20 Automorphisms and endomorphisms
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[1] DOI: 10.1080/00927878008822456 · Zbl 0423.20005 · doi:10.1080/00927878008822456
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