On \(U\)-codominant dimension (Q6622153)

For a ring \(R\) and a given left (or right) \(R\)-module \(U\), the authors in the paper under review introduce the notion of the \(U\)-codominant dimension of a left (or right) \(R\)-module \(M\), which is a dual notion of the \(U\)-dominant dimension introduced by \textit{T. Kato} [Tôhoku Math. J. (2) 21, 321--327 (1969; Zbl 0191.04002)]. Given two semiregular rings \(R\) and \(S\), and a semidualizing \((R,S)\)-bimodule \(U\), the authors prove that the \(U\)-codominant dimension of \({_R}U\) and \(U_S\) are identical. As an application, the authors prove that the \(U\)-codominant dimension of \(U\) is at least two if and only if the functor \(U\otimes_S\mathrm{Hom}_R(U,-)\) is right exact and if and only if the functor \(\mathrm{Hom}_R(U,U\otimes_S-)\) is left exact.