Relative Projective Dimensions

In (n, d)-ring and n-coherent ring theory, n-presented modules plays an important role. In this paper, we firstly give some new characterizations of n-presented modules and n-coherent rings. Then, we introduce the concept of (n,0)-projective dimension, which measures how far away a finitely generated module is from being n-presented and how far away a ring is from being Noetherian, for modules and rings. This dimension has nice properties when the ring in question is n-coherent. Some known results are extended or obtained as corollaries.