Steady ideals and rings (Q1378842)

The authors call a right \(R\)-module \(M\) dually slender, if the hom functor \(\Hom_R(M,-)\) commutes with direct sums. A ring \(R\) is called right steady, if the finitely generated right \(R\)-modules are the only dually slender right \(R\)-modules. The authors look for conditions under which the rings are steady, under the assumption that all the two-sided ideals are countably generated. For instance, if \(R\) is an abelian regular ring such that all ideals are countably generated, then \(R\) is steady. Moreover a regular ring with all ideals countably generated and all primitive factor-rings artinian is right steady. A commutative chain ring is steady iff \(\text{rad}(R)\) is countably generated and the ring \(R/\text{rad}(R)\) contains no uncountable chains of ideals (\(\text{rad}(R)\) denotes the prime radical). Utilizing a result of Jategaonkar, an example is produced to show that right steady need not imply left steady property.