Neat-phantom and clean-cophantom morphisms

Abstract

Let [Formula: see text] be the class of all left [Formula: see text]-modules [Formula: see text] which has a projective resolution by finitely generated projectives. An exact sequence [Formula: see text] of right [Formula: see text]-modules is called neat if the sequence [Formula: see text] is exact for any [Formula: see text]. An exact sequence [Formula: see text] of left [Formula: see text]-modules is called clean if the sequence [Formula: see text] is exact for any [Formula: see text]. We prove that every [Formula: see text]-module has a clean-projective precover and a neat-injective envelope. A morphism [Formula: see text] of right [Formula: see text]-modules is called a neat-phantom morphism if [Formula: see text] for any [Formula: see text]. A morphism [Formula: see text] of left [Formula: see text]-modules is said to be a clean-cophantom morphism if [Formula: see text] for any [Formula: see text]. We establish the relationship between neat-phantom (respectively, clean-cophantom) morphisms and neat (respectively, clean) exact sequences. Also, we prove that every [Formula: see text]-module has a neat-phantom cover with kernel neat-injective and a clean-cophantom preenvelope with cokernel clean-projective.