关于相对平坦性与相对凝聚性的注记

Let $R$ be a ring and $M$ a fixed right $R$-module. A new characterization of $M$-flatness is given by certain linear equations. For a left $R$-module $F$ such that the canonical map $M\otimes_RF\rightarrow \mathrm{Hom}_R(M^*, F)$ is injective, where $M^* =$ Hom$_R(M, R)$, the $M$-flatness of $F$ is characterized via certain matrix subgroups. An example is given to show that $R$ need not be $M$-coherent even if every left $R$-module is $M$-flat. Moreover, some properties of $M$-coherent rings are discussed.