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In Lawrence, John. "A countable self-injective ring is quasi-Frobenius." Proceedings of the American Mathematical Society (1977): 217-220. the first line is this:

It has been known for some time that a countable self-injective ring is semilocal (see for example [8].)

Where [8] is Renault, Guy. "Sur les anneaux de groupes." CR Acad. Sci. Paris Sér. AB 273 (1971): A84-A87.

The problem is that Renault's paper, as far as I know, says nothing of the sort. There is a result about semilocal group rings (which does not mention self-injectivity or countability) and a result on self-injective group rings (which proves the group ring must be semilocal) but nothing in general about "countable self-injective rings."

I'm also pretty familiar with everything else cited in Lawrence's paper, and I don't think it was a case of the wrong bibliography item being cited. I can see is that a method like the one used by Gentile will prove that a countable, right self-injective ring is semi local, although I can’t see that it is explicit in Gentile’s paper.

Is anyone aware of a reference that proves such a thing in the box above? It could perhaps be a different Renault paper.

Or maybe you can explain how that statement actually follows from Renault's paper. I prepared a translation of it so I thought I should immediately see where that might be explained, but who knows: maybe I'm overlooking something obvious.

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    $\begingroup$ You could, of course, contact John Lawrence. He is retired, but still active mathematically. $\endgroup$
    – David Handelman
    Commented Jun 10, 2018 at 13:54
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    $\begingroup$ @DavidHandelman Ok, I'll try that too. Quite often when I ask someone "what did you mean when you wrote this 40 years ago" the answer is indeterminate, but it's worth a shot :) $\endgroup$
    – rschwieb
    Commented Jun 10, 2018 at 15:30
  • $\begingroup$ @rschwieb As far as I could see, John Lawrence's proof that countable, self-injective rings are quasi-Frobenius does not use his claim that such rings are semilocal. But quasi-Frobenius rings are semilocal. So, doesn't his paper give a proof of the claim in the box? $\endgroup$
    – Pace Nielsen
    Commented Apr 24 at 16:10
  • $\begingroup$ @PaceNielsen Sure: the question asks for what other publications Lawrence could have been referring to, since it is not in his paper and seemingly not in the paper listed as the reference. At worst, it is folklore somehow. IIRC I fabricated a proof in an article of mine based on ideas from Gentile. $\endgroup$
    – rschwieb
    Commented Apr 24 at 17:41
  • $\begingroup$ @rschwieb The only other thought I had was to contact an expert on self-injectivity, like S.K. Jain, to see if he knows anything about this. $\endgroup$
    – Pace Nielsen
    Commented Apr 30 at 4:32

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