TITLE: Is there a notion of 'amenable ring'
QUESTION [12 upvotes]: Amenable groups are everywhere these days, as examples of all kinds of lovely phenomena. And there are various ways of defining notions of 'amenable monoid' or possibly 'amenable semigroup'. But for monoids and semigroups, amenability conditions that are known to be equivalent for groups are not always equivalent and there is a more complicated story unfolding. Nevertheless it seems natural to ask if there is a notion for rings.

REPLY [15 votes]: Yes, see Elek, Gábor
The amenability of affine algebras. 
J. Algebra 264 (2003), no. 2, 469–478. 
and Gromov, Misha Entropy and isoperimetry for linear and non-linear group actions. Groups Geom. Dyn. 2 (2008), no. 4, 499-–593.