6
$\begingroup$

I'm seeking a book or a survey providing an overview, as rich as possible, of the literature on Dedekind-finite (or von Neumann-finite) rings (let me recall that a unital ring $R$ is Dedekind-finite if $xy = 1_R$ for some $x, y \in R$ implies $yx = 1_R$).

There is something in C. Faith's Algebra II - Ring Theory (1979) and other classical monographs on rings and modules, but they are not very up to date. On the other hand, Y.T. Lam's A First Course in Noncommutative Rings (2nd ed., 2001) and Lectures on Modules and Rings (1999) include more recent developments, but many significant results are left as exercises for the reader, and this is not what I'm looking for.

Improve this question
$\endgroup$
5
  • $\begingroup$ I doubt there's a single reference, but you might also search {\it stably finite,} [all matrix rings are D-finite] which plays a significant role in C*-algebras and von Neumann regular rings (for the latter, there is the book by Goodearl, {\em Von Neumann regular rings.} And another term (which I prefer) for Dedekind finite is {\it directly finite.} $\endgroup$
    – David Handelman
    Commented Feb 28, 2017 at 14:21
  • 1
    $\begingroup$ There are so many papers which make use of stable finiteness (especially in C*-algebras and regular rings) that it really depends on what you want. If I can plug one of my earliest papers, Rank functions and K0 of regular rings K.R. Goodearl & D. Handelman, J Pure & Applied Algebra 7 (1976), 195-216. $\endgroup$
    – David Handelman
    Commented Feb 28, 2017 at 14:34
  • $\begingroup$ Thanks for the helpful pointers. Actually, it would be already enough for me to find a book or a paper giving a list of (as many as possible) natural examples (apart from the trivial ones), either with proofs or with punctual references. The list should include at least left (resp., right) Noetherian rings, rings with finitely many nilpotents, algebraic algebras over a field, and PI-algebras. Does this make any difference for my request? $\endgroup$
    – Salvo Tringali
    Commented Feb 28, 2017 at 14:35
  • $\begingroup$ The order of the last two comments was reversed, because mine was a response to Salvo's, which was modified while I was typing it, ... $\endgroup$
    – David Handelman
    Commented Feb 28, 2017 at 14:53
  • 2
    $\begingroup$ If your issue with Lam is that he leaves things to the exercise, note that he works out every exercise in his exercise books. So you could read the solutions as needed. $\endgroup$
    – Pace Nielsen
    Commented Mar 16, 2017 at 13:55

0

Reset to default

You must log in to answer this question.