Characterizing almost perfect rings by covers and envelopes
J. Korean Math. Soc.
Published online September 3, 2019
Laszlo Fuchs
Tulane University
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}, who showed 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 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 domains are adapted to rings with zero-divisors.
Keywords : Cotorsion Cotorsion cotorsion pairs, covers and envelopes; strongly flat, weak-injective modules; perfect, subperfect, almost perfect rings perfect, subperfect, almost perfect rings
MSC numbers : Primary 13C05. Secondary 13C11, 13F05
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