Given a multiplicative subset S in a commutative ring R, we consider S-weakly cotorsion and S-strongly flat R-modules, and show that all R-modules have S-strongly flat covers if and only if all flat R-modules are S-strongly flat. These equivalent conditions hold if and only if the localization R_S is a perfect ring and, for every element s ∈ S, the quotient ring R/sR is a perfect ring, too. The multiplicative subset S ⊂ R is allowed to contain zero-divisors.
S-almost perfect commutative rings
Bazzoni S.
;
2019
Abstract
Given a multiplicative subset S in a commutative ring R, we consider S-weakly cotorsion and S-strongly flat R-modules, and show that all R-modules have S-strongly flat covers if and only if all flat R-modules are S-strongly flat. These equivalent conditions hold if and only if the localization R_S is a perfect ring and, for every element s ∈ S, the quotient ring R/sR is a perfect ring, too. The multiplicative subset S ⊂ R is allowed to contain zero-divisors.
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