A Dedekind-finite ring is a ring in which $ab=1$ implies $ba=1$.

It seems natural to look for a connection to Dedekind-finite sets, however for such a set any injective endomorphism is surjective, while for a Dedekind-finite ring it goes vice versa, namely, any surjective endomorphism is injective (In other words, such a ring is Hopfian).

So, what is the motivation behind this name (for rings)?



It would seem to me that you should simply apply your own observation concerning Dedekind-finite sets and their definition to the left/right homotheties involved: given $ab=1$, the right homothety defined by $b$, i.e. $\vartheta_b:R\rightarrow R$, $r\mapsto rb$, viewed as a homomorphism of abelian groups, say, is clearly surjective (one has $\vartheta_b\circ\vartheta_a=\text{id}_R$); iff also $ba=1$, then $\vartheta_a\circ\vartheta_b=\text{id}_R$, making $\vartheta_b$ injective, too (note also that the one-sided multiplicative inverses of an element, when they exist, must coincide due to associativity). The endomorphism rings of finite-dimensional vector spaces over (skew) fields are, of course, standard examples of Dedekind-finite rings, further justifying (possibly) the intuitive feel that such vector spaces (and hence their endomorphism rings) are "small" in a sense. Kind regards, Stephan F. Kroneck.


Lam has an exercise on this in Lectures on Modules and Rings pp 18:

A module M is called Dedekind finite if $M\cong M\oplus N$ implies $N=0$. $M$ is a Dedekind finite module iff $End(M_R)$ is a Dedekind finite ring. If $M$ is Hopfian, then $End(M_R)$ is Dedekind finite, but not always conversely. The case when $M=R_R$, Dedekind finiteness of $R_R$ turns out to be equivalent to being a Hopfian module, since $R_R$ is projective.

I spent some time looking at rings where $R_R$ was $\textit{coHopfian}$, and found some interesting stuff. For one thing, it's not the same as being a coHopfian object in the category of rings. It took a lot of digging but I finally found an example given by Varadarajan of a left-not-right (module)-coHopfian ring.

  • $\begingroup$ @BillDubuque We are definitely not on those terms yet, but as a compromise, you can find me idling in the crusade of answers chat during the day. $\endgroup$ – rschwieb Feb 6 '14 at 11:34
  • $\begingroup$ @BillDubuque Let's at least try chat first, and go from there. $\endgroup$ – rschwieb Feb 11 '14 at 19:58
  • $\begingroup$ @BillDubuque We are definitely not on those terms yet, but as a compromise, you can find me idling in the crusade of answers chat during the day. Even if I'm not logged in, you can click my name and initiate a semiprivate chat. I don't think it's a very good idea to engage in private correspondence with anyone I don't know well, or who engages in less-than-friendly interactions with me. You could begin to dispel that impression by engaging in casual chats as I am suggesting. $\endgroup$ – rschwieb Apr 22 '14 at 0:14
  • $\begingroup$ @BillDubuque This is all just commonsense internet privacy practice. I wouldn't give you my home address either. If you ever get over this strange aversion to a simple chat, just let me know. $\endgroup$ – rschwieb Apr 22 '14 at 2:30
  • $\begingroup$ @BillDubuque trust me, I will not be going that far out of my way at this time when you won't even take me up on this harmless chat compromise. I could change my mind someday, if forthcoming conversations aren't so adversarial, but not this month. Good evening to you. $\endgroup$ – rschwieb Apr 22 '14 at 2:45

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.