Abstract
From the early 1960s to the early 1970s, there was much activity leading up to a proof that if the group ring R[G] is right self-injective, then G is finite. Unfortunately, it is difficult to find this fact both stated and proven in print and in full generality. This article presents new renditions of the main proofs, and chronicles what the author learned while sorting out the literature for this period.
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Notes
The author would like to thank Keith A. Kearnes for illuminating how this lemma works.
Another echo from the proof of Pascaud’s lemma.
For those not familiar with this lemma: by injectivity, \(\tau\) extends to the R[H] linear homomorphism \({{\bar{\tau }}}:R[H]\rightarrow R[H]\), and then \({{\bar{\tau }}}(1)\) is the d we seek.
See [5] for the characterization of when a group ring is right hereditary.
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Acknowledgements
The author would like to thank the referee and Greg Oman for their helpful feedback on earlier drafts. Special thanks also goes to several of the original authors, in particular Ian Connell, Daniel Farkas, Surender K. Jain, Gordon Mason, and Donald Passman, who were kind enough to correspond about the era of these theorems.
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Communicated by Francisco César Polcino Milies.
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Schwiebert, R.C. Results related to self-injectivity of the group ring. São Paulo J. Math. Sci. 14, 651–664 (2020). https://doi.org/10.1007/s40863-020-00176-2
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DOI: https://doi.org/10.1007/s40863-020-00176-2