Homological properties of modules characterized by matrices

摘要

Some homological properties of R-modules were investigated by matrices over a ring R. Given two cardinal numbers α, β and an α x β row-finite matrix A, it was proved that Ext_R~1(R~((α))/R~((β))A, M) = 0 if and only if M_α/r_(M_α)(R~((β))A) ≈ Hom_R(R~((β))A,M) if and only if r_(M_β)l_(R~((β)))(A) = AM_α. Thus, the notion of (m,n)-injectivity was extended. Moreover, ( α, β) -flatness was characterized via annihilators of matrices, factorizations of homomorphisms as well as homological groups so that (m, n)-flat modules, f-projective modules and n-projective modules were consolidated under the notion of (α, β)-flat modules. Furthermore, a characterization of left R-ML modules and some equivalent conditions for R~((β)) to be left R-ML were presented. Consequently, the notions of coherent rings, (m, n)-coherent rings and π-coherent rings were consolidated under that of (α, β)-coherent rings.