Fp - Projective Dimensions

Let R be a ring and M a right R-module. Ng (1984) defined the finitely presented dimension f p dim M of M as inf n there exists an exact sequence Pn+1 → Pn → · · · → P0 → M → 0 of right R-modules, where each Pi is projective, and Pn+1 Pn are finitely generated . If no such sequence exists for any n, set f p dim M = . The right finitely presented dimension r f p dim R of R is defined as sup f p dim M M is a finitely generated right R-module . The dimension defined in this way has some nice properties, but no ring or finitely generated module can have finitely presented dimension 1 by Ng (1984), Proposition 1.5 and Corollary 1.6. To fill the gap, we shall introduce another kind of finitely presented dimension of modules and rings in this paper. In Section 2, the definition and some general results are given. For a right Rmodule M , we define the FP-projective dimension fpd M of M to be the smallest integer n ≥ 0 such that Extn+1 M N = 0 for any FP-injective right R-module N . If no such n exists, set fpd M = . The right FP-projective dimension rfpD R of a ring R is defined as sup fpd M M is a finitely generated right R-module . M is called FP-projective if fpd M = 0, i.e., Ext M N = 0 for any FP-injective right