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| Let R be a ring and A an $\beta\times\alpha$ matrix over R, where $\alpha$ and $\beta$ are two card numbers. We investigate some homological properties of modules over R by matrices. Thus, $(\alpha, \beta)$-injective (resp., $(\alpha, \beta)$-flat) modules and $(\alpha, \beta)$-coherent rings are introduced and investigated. Some known results on generalized injective (resp., projective, flat) modules as well as ($\pi$-)coherent rings are extended. |
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| Keywords:$(\alpha, \beta)$-injective module; $(\alpha, \beta)$-flat module; R-ML module; $(\alpha, \beta)$-coherent ring |
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