Classes of modules closed under projective covers

Abstract In this work, we study some classes of modules closed under submodules, quotients, and projective covers, even if the left projective cover of an arbitrary left module not always exists. We obtain a characterization of artinian principal ideal rings when the class of left <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>R</m:mi> </m:math> R -modules closed under submodules and projective covers and the class of left <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>R</m:mi> </m:math> R -modules closed under quotients and projective covers coincide. Also, for commutative rings, we characterize noetherian rings in which every cyclic module is quasi-injective.