Mehrvarz, A. A.; Wallace, D. A. R. On the automorphisms of the group ring of a unique product group. (English) Zbl 0582.16002 Proc. Edinb. Math. Soc., II. Ser. 30, 201-205 (1987). 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. Cited in 1 ReviewCited in 1 Document MSC: 16S34 Group rings 16W20 Automorphisms and endomorphisms Keywords:nilpotent ideal; group ring; R-linear augmentation preserving automorphism; unique product group PDFBibTeX XMLCite \textit{A. A. Mehrvarz} and \textit{D. A. R. Wallace}, Proc. Edinb. Math. Soc., II. Ser. 30, 201--205 (1987; Zbl 0582.16002) Full Text: DOI References: [1] DOI: 10.1080/00927878008822456 · Zbl 0423.20005 · doi:10.1080/00927878008822456 [2] Sehgal, Topics in Group Rings (1978) · Zbl 0411.16004 [3] Passman, The Algebraic Structure of Group Rings (1977) · Zbl 0368.16003 [4] Brodskh, Uspekhi. Mat. Nauk 35 pp 183– (1980) [5] DOI: 10.1080/00927878108822598 · Zbl 0455.20025 · doi:10.1080/00927878108822598 [6] DOI: 10.1007/BF01214717 · Zbl 0471.20017 · doi:10.1007/BF01214717 [7] DOI: 10.1112/plms/s2-46.1.231 · Zbl 0025.24302 · doi:10.1112/plms/s2-46.1.231 [8] Parmenter, Canad. Math. Bull. 18 pp 567– (1975) · Zbl 0331.16009 · doi:10.4153/CMB-1975-101-3 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.