While (once again) perusing T.Y. Lam's excellent GTM 189 "Lectures on Modules and Rings" I compared the various conditions given that typically imply IBN, e.g. the (left) strong rank condition, stably finite etc. One condition on a ring $R$, even stronger than both of the aforesaid, would be that all the (left) finitely generated free modules over $R$ are cohopfian. For instance, this is clearly the case, and in a strong form, when $R$ is (left) artinian (and hence is not completely without interest; free modules over such rings satisfy just about all the important fundamental properties of vector spaces over skew fields). My question: have such rings been graced with a name, and is the notion left/right independent or not ? (The term "strongly finite" or such like, I believe, is used by functional analysts in a different sense.) Thanks in advance for any tips ! Kind regards, Stephan.
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$\begingroup$ What is cohopfian? I understand that all fg projectives are hopfian is the same as stably finite. $\endgroup$– David HandelmanApr 22, 2017 at 14:43
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$\begingroup$ Definition of IBN: en.wikipedia.org/wiki/Invariant_basis_number $\endgroup$– YCorApr 22, 2017 at 14:45
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1$\begingroup$ If $R/J(R)$ is von Neumann regular and stably finite, then every one to one matrix is invertible, hence $R$ is cohopfian (I had to look up the definition). The converse is false (or anyway, I'm pretty sure it's false). $\endgroup$– David HandelmanApr 22, 2017 at 14:52
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