Results related to self-injectivity of the group ring
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Results related to self‑injectivity of the group ring Ryan C. Schwiebert1,2
© Instituto de Matemática e Estatística da Universidade de São Paulo 2020
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. Keywords Self-injective ring · Group ring · Group algebra Mathematics Subject Classification Primary 16S34 · Secondary 16D50
1 Introduction For a ring R and a group G, the group ring R[G] is of enormous interest to group representation theory and ring theory. R[G] can be defined as the set of formal lin∑ ear combinations g∈G rg g where the rg ∈ R and only finitely many of which are nonzero, imbued with the following operations: (∑ ) (∑ ) ∑ rg g + r � gg = (rg + rg� )g
(∑
rg g
)(∑
) ∑∑ r � gg = (rh rk� )g g∈G hk=g
Communicated by Francisco César Polcino Milies. * Ryan C. Schwiebert [email protected] 1
Seegrid Corp., 216 RIDC Park W Dr, Pittsburgh, PA 15275, USA
2
Center of Ring Theory and Its Applications, Ohio University, 321 Morton Hall, Athens, OH 45701, USA
13
Vol.:(0123456789)
São Paulo Journal of Mathematical Sciences
For ring theorists, it is very interesting to study how properties are or are not preserved between R and R[G]. The results are sometimes hard to anticipate. For instance, consider the following three results [4, Theorems 1,2,3, pp. 657–660] (1) R[G] is a right Artinian ring if and only if R is right Artinian and G is finite. (2) R[G] is right Noetherian if R is right Noetherian and G is finite, but the converse is not true. (3) R[G] is von Neumann regular if and only if R is von Neumann regular and G is a locally finite group such that the order of each element is a unit in R. The topic of this article is a theorem of the sort described above (or rather, one direction of the theorem): Theorem 1 For a ring R and a group G, the group ring R[G] is right self-injective if and only if R is right self-injective and G is finite. In the last half of the twentieth century, complete proofs of the sufficiency direction of Theorem 1 and the portion of the necessity direction about R being right self-injective were readily available and well-referenced. By the early 1970s it was well-known that G must be locally finite, but the most elusive portion was to show that if R[G] is right self-injective, then G is finite. A researcher following the trails of citations will encounter some difficulties when trying to locate a proof. The author was only able to locate two sources with statements and proofs of the statement of Theorem 1. One is Renault’s paper in French, which requires the reader to unravel a few overly general citations. The other is [25] (or its Russian original); however, the author did not encounter any r
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