Abstract:In this paper, we show the existence of copure injective preenvelopes over noetherian rings and copure flat preenvelopes over commutative artinian rings. We use this to characterize $n$-Gorenstein rings. As a consequence, if the full subcategory of strongly copure injective (respectively flat) modules over a left and right noetherian ring $R$ has cokernels (respectively kernels), then $R$ is $2$-Gorenstein.
Keywords: preenvelopes, copure injective, copure flat, $n$-Gorenstein, resolutions
AMS Subject Classification: 18G05, 18G10, 18G15