Flat modules and coherent endomorphism rings relative to some matrices

Let $N$ be a left $R$-module with the endomorphism ring $S=\text{End}{(}_{R}N)$. Given two cardinal numbers $\alpha$ and $\beta$ and a matrix $A\in S^{\beta \times \alpha }$, $N$ is called flat relative to $A$ in case, for each $x\in l_{N^{(\beta )}}(A)=\{u\in N^{(\beta )}\mid uA=0\}$, there are a positive integer $k$, $y\in N^k$ and a $k\times \beta$ row-finite matrix $C$ over $S$ such that $CA=0$ and $x=yC$. It is shown that $N_S$ is flat relative to a matrix $A$ if and only if $l_{N^{(\beta )}}(A)$ is generated by $N$. $S$ is called left coherent relative to $A$ if Ker ${(}_S{S}^{(\beta )}{\to }_S{S}^{(\beta )}A)$ is finitely generated. It is shown that $S$ is left coherent relative to $A$ if and only if Hom ${}_R(N,l_{N^n}(A))$ is a finitely generated left $S$-module if and only if $l_{N^n}(A)$ has an add $(N)$-precover (add $(N)$ denotes the category of all direct summands of finite direct sums of copies of ${}_{R}N$). Regarding applications, new necessary and sufficient conditions for epic (monic, having the unique mapping property) add $(N)$-precovers of $l_{N^{(\beta )}}(A)$ are investigated. Also, some new characterizations of left $n$-semihereditary rings and von Neumann regular rings are given.