On (, U)-Coherence of Modules and Rings

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 [M] is (, U)-finitely presented in [M]. It is proved under some additional conditions that a left R-module M is (, U)-coherent if and only if i1U 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.