On envelopes with the unique mapping property

摘要

We prove that (a) if R is a left coherent ring, then the weak global dimension w D(R) = n (n = 2) if and only if every (n #x2013; 2)th F#x2013;cosyzygy of a finitely presented right R#x2013;module has a flat envelope with the unique mapping property; (b) if R is a left coherent and right perfect ring, then the right global dimension rD(R) = n (n = 2) if and only if every (n #x2013; 2)th P#x2013;cosyzygy of a right R#x2013;module has a projective envelope with the unique mapping property; (c) if R is a commutative ring, then R is #x3C0;#x2014;coherent (resp. coherent) and the exactness of 0 - K - F0- F1with Foand F1(finitely) projective and K finitely generated implies the projectivity of K if and only if every finitely generated (resp, finitely presented) R#x2013;module has a (finitely) projective envelope with the unique mapping property.