In the ring theory, a left perfect ring is a type of ring in which all left modules have projective covers. The right case is defined by analogy and a ring is perfect means it is both left and right perfect. You can see more at wiki perfect ring.
But in the commutative algebra, we have a similar concept perfect module. A R-module M is perfect, if proj dim(M)=grade(M). But I do not find the word perfect module here in the wiki.
Is any relationship between the perfect ring in the ring theory and the perfect module in the commutative algebra?
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$\begingroup$In my experience, there is a definition of "perfect module" that aligns with perfect rings is the one given here in Wisbauer's text.
We call $N$ perfect in $σ[M]$ if, for every index set $Λ$, the sum $N^{(Λ)}$ is semiperfect in $σ[M]$.
In the category of $R$ modules, that would be
We call the $R$ module $N$ perfect if, for every index set $Λ$, the sum $N^{(Λ)}$ is a semiperfect $R$ module$.
I do not know if this relates to the commutative algebra definition, but on the surface it does not look like it. It could just be a case of divergence in terminology due to the choice of generic adjectives like regular, normal, perfect which show up a lot.
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