Some homological properties of R-modules were investigated by matrices over aring R. Given two cardinal numbers α, β and an α x β row-finite matrix A, it was proved thatExt_R^1(R^((α))/R^((β))A, M) = 0 if and on...Some homological properties of R-modules were investigated by matrices over aring R. Given two cardinal numbers α, β and an α x β row-finite matrix A, it was proved thatExt_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) ifand 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 ofhomomorphisms as well as homological groups so that (m, n)-flat modules, f-projective modules andn-projective modules were consolidated under the notion of (α, β)-flat modules. Furthermore, acharacterization of left R-ML modules and some equivalent conditions for R^((β)) to be left R-MLwere presented. Consequently, the notions of coherent rings, (m, n)-coherent rings and π-coherentrings were consolidated under that of (α, β)-coherent rings.展开更多
文摘Some homological properties of R-modules were investigated by matrices over aring R. Given two cardinal numbers α, β and an α x β row-finite matrix A, it was proved thatExt_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) ifand 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 ofhomomorphisms as well as homological groups so that (m, n)-flat modules, f-projective modules andn-projective modules were consolidated under the notion of (α, β)-flat modules. Furthermore, acharacterization of left R-ML modules and some equivalent conditions for R^((β)) to be left R-MLwere presented. Consequently, the notions of coherent rings, (m, n)-coherent rings and π-coherentrings were consolidated under that of (α, β)-coherent rings.
