Strong incompactness for some nonperfect rings

Abstract:

Answering a question of Eklof and Mekler, for each regular uncountable cardinal $\kappa$, we construct a non-left-perfect ring ${R_\kappa }$ and a non-projective strongly $\kappa$-free left ideal ${I_\kappa }$ such that ${\text {gen}}({I_\kappa }) = \kappa$. Moreover, if $\kappa > {\aleph _1}$, then ${I_\kappa }$ is not $\kappa$-free. As consequences, we obtain results concerning incompactness spectra of non-perfect rings.