Characterizing almost perfect rings by covers and envelopes
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Characterizations of almost perfect domains by certain covers and envelopes, due to Bazzoni--Salce \cite{BS1} and Bazzoni \cite{B}, are generalized to almost perfect commutative rings (with zero-divisors). These rings were introduced recently by Fuchs--Salce \cite{FS}, showing that the new rings share numerous properties of the domain case. In this note, it is proved that admitting strongly flat covers characterizes the almost perfect rings within the class of commutative rings (Theorem \ref{Ce}). Also, the existence of projective dimension 1 covers characterizes the same class of rings within the class of commutative rings admitting the cotorsion pair $(\PP_1, \DD)$ (Theorem \ref{Fd}). Similar characterization is proved concerning the existence of divisible envelopes for $h$-local rings in the same class (Theorem \ref{Le}). In addition, Bazzoni's characterization {\it via} direct sums of weak-injective modules \cite{B} is extended to all commutative rings (Theorem \ref{De}). Several ideas of the proofs known for integral domains are adapted to rings with zero-divisors.
Abstract
Characterizations of almost perfect domains by certain covers and envelopes, due to Bazzoni--Salce \cite{BS1} and Bazzoni \cite{B}, are generalized to almost perfect commutative rings (with zero-divisors). These rings were introduced recently by Fuchs--Salce \cite{FS}, showing that the new rings share numerous properties of the domain case. In this note, it is proved that admitting strongly flat covers characterizes the almost perfect rings within the class of commutative rings (Theorem \ref{Ce}). Also, the existence of projective dimension 1 covers characterizes the same class of rings within the class of commutative rings admitting the cotorsion pair $(\PP_1, \DD)$ (Theorem \ref{Fd}). Similar characterization is proved concerning the existence of divisible envelopes for $h$-local rings in the same class (Theorem \ref{Le}). In addition, Bazzoni's characterization {\it via} direct sums of weak-injective modules \cite{B} is extended to all commutative rings (Theorem \ref{De}). Several ideas of the proofs known for integral domains are adapted to rings with zero-divisors.
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