On co-Noetherian dimension of rings. (Q1954009)

A right module \(M_R\) over a ring \(R\) with identity element is finitely cogenerated (originally called ``finitely embedded'' by \textit{P.~Vámos} [J. Lond. Math. Soc. 43, 643-646 (1968; Zbl 0164.04003)]), if its injective hull is of the form \(\text{E}(M)=\sum_{i=1}^n\text{E}(S_i)\), where each \(S_i\) is a simple \(R\)-module. \textit{J. P.~Jans} [J. Lond. Math. Soc., II. Ser. 1, 588-590 (1969; Zbl 0185.09203)] called a ring \(R\) co-Noetherian if factors of finitely cogenerated right \(R\)-modules are finitely cogenerated, and he proved that this condition is equivalent to finitely cogenerated \(R\)-modules being Artinian. Motivated by this result, the authors define the co-Noetherian dimension of \(R\) to be the least nonnegative integer \(n\) such that the Krull dimension (in the sense of Gabriel and Rentschler) of all finitely cogenerated right \(R\)-modules is \(\leq n\). It is shown that for each integer \(n\geq 1\) there exists a ring \(R\) with co-Noetherian dimension \(\text{co-N.dim}(R)=n\). Also, if \(R\) is commutative and the injective hulls of all simple \(R\)-modules have Krull dimension, then \(\text{co-N.dim}(R)\) equals the supremum of the Krull dimensions of certain prime factor rings of \(R\). Furthermore, if both the co-Noetherian and the Krull dimensions of \(R\) exist, then the former is less than or equal to the latter.