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Copure injective resolutions, flat resolvents and dimensions

Edgar E. Enochs; Jenda M. G. Overtoun — 1993

Commentationes Mathematicae Universitatis Carolinae

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.

On Cohen-Macaulay rings

Edgar E. Enochs; Jenda M. G. Overtoun — 1994

Commentationes Mathematicae Universitatis Carolinae

In this paper, we use a characterization of $R$ -modules $N$ such that $f d_{R} N = p d_{R} N$ to characterize Cohen-Macaulay rings in terms of various dimensions. This is done by setting $N$ to be the $d t h$ local cohomology functor of $R$ with respect to the maximal ideal where $d$ is the Krull dimension of $R$ .