Steadiness of polynomial rings

Abstract

A module \(M\) is said to be small if the functor \(\operatorname{Hom}(M,-)\) commutes with direct sums and right steady rings are exactly those rings whose small modules are necessary finitely generated. We give several results on steadiness of polynomial rings, namely we prove that polynomials over a right perfect ring such that \(End_R(S)\) is finitely generated over its center for every simple module \(S\) form a right steady ring iff the set of variables is countable. Moreover, every polynomial ring in uncountably many variables is non-steady.