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Homological properties of modules characterized by matrix Xiaoxiang Zhang and Jianlong Chen Department of Mathematics, Southeast University Nanjing, PRC 210096 z990303@ Abstract Let R be a ring and A an β×α matrix over R , where α and β are two card numbers. We investigate some homological properties of modules over R by matrices. Thus, (α, β)-inj ective ( resp., (α, β)-flat ) modules and (α, β)-coherent rings are introduced and investigated. Some known results on generalized inj ective (resp., projective, flat) modules as well as (π-)coherent rings are extended. Keywords: (α,β)-injective mo dule; (α,β)-fla t module; R-ML module; (α,β)-coherent ring 1 Introduction In 2001, (m, n)-inj ective modules were discussed by Chen, Ding, Li and Zhou[1]. Given m and n ∈N (the set of natural numbers), a right R-module M is called (m, n)-injective if every right R-homomorphism m m from an n-generated submodule of R to M extends to one from R to M . This definition unifies several definitions on generalizations of inject ivity of modules, such as f-inj ective modules, P-inject ive modules and FP-injective modules. Many characterizations of (m, n)-injective modules were given and the notions [2] of (m, n)-flat modules as well as (m, n)-coherent rings were introduced b