@article{Zeng2023, 
author = {Yuedi Zeng},
title = {Flat modules and coherent endomorphism rings relative to some matrices},
year = {2023},
journal = {AIMS Mathematics},
volume = {8},
number = {6},
pages = {14111-14131},
keywords = {coherent, flat, endomorphism ring, precover, quasi-injective},
url = {https://www.sciopen.com/article/10.3934/math.2023721},
doi = {10.3934/math.2023721},
abstract = {Let    N be a left    R-module with the endomorphism ring    S  =  End      (          R        N  ). Given two cardinal numbers    α and    β and a matrix    A  ∈      S          β      ×      α      ,    N is called flat relative to    A in case, for each    x  ∈      l                  N                  (          β          )                      (  A  )  =  {  u  ∈      N          (      β      )        ∣  u  A  =  0  }, there are a positive integer    k,    y  ∈      N          k       and a    k  ×  β row-finite matrix    C over    S such that    C  A  =  0 and    x  =  y  C. It is shown that        N          S       is flat relative to a matrix    A if and only if        l                  N                  (          β          )                      (  A  ) is generated by    N.    S is called left coherent relative to    A if Ker       (          S            S          (      β      )            →          S            S          (      β      )        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                  (          β          )                      (  A  ) are investigated. Also, some new characterizations of left    n-semihereditary rings and von Neumann regular rings are given.}
}