Let U be a flat right R-module and ? an infinite cardinal number. A left R-module M is said to be (?, U)-coherent if every finitely generated submodule of every finitely generated M-projective module in σ[M] is (?, U...Let U be a flat right R-module and ? an infinite cardinal number. A left R-module M is said to be (?, U)-coherent if every finitely generated submodule of every finitely generated M-projective module in σ[M] is (?, U)-finitely presented in σ[M]. It is proved under some additional conditions that a left R-module M is (?, U)-coherent if and only if is M-flat as a right R-module if and only if the (?, U)-coherent dimension of M is equal to zero. We also give some characterizations of left (?, U)-coherent dimension of rings and show that the left ?-coherent dimension of a ring R is the supremum of (?, U)-coherent dimensions of R for all flat right R-modules U.展开更多
基金Research supported by National Natural Science Foundation of China(10171082)by the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of MOE,P.R.C.
文摘Let U be a flat right R-module and ? an infinite cardinal number. A left R-module M is said to be (?, U)-coherent if every finitely generated submodule of every finitely generated M-projective module in σ[M] is (?, U)-finitely presented in σ[M]. It is proved under some additional conditions that a left R-module M is (?, U)-coherent if and only if is M-flat as a right R-module if and only if the (?, U)-coherent dimension of M is equal to zero. We also give some characterizations of left (?, U)-coherent dimension of rings and show that the left ?-coherent dimension of a ring R is the supremum of (?, U)-coherent dimensions of R for all flat right R-modules U.
