@article{58182,
  abstract     = {{We study a weak divisibility property for noncommutative rings: a nontrivial ring is fadelian if for all nonzero a and x there exist b, c such that x=ab+ca. We prove properties of fadelian rings and construct examples thereof which are not division rings, as well as non-Noetherian and non-Ore examples.}},
  author       = {{Khanfir, Robin and Seguin, Beranger Fabrice}},
  issn         = {{0219-4988}},
  journal      = {{Journal of Algebra and Its Applications}},
  publisher    = {{World Scientific Pub Co Pte Ltd}},
  title        = {{{Study of a division-like property}}},
  doi          = {{10.1142/s0219498825502214}},
  year         = {{2024}},
}

@article{33262,
  abstract     = {{The authors of Berg et al. [J. Algebra 348 (2011) 446–461] provide an algorithm for finding a complete system of primitive orthogonal idempotents for CM, where M is any finite R-trivial monoid. Their method relies on a technical result stating that R-trivial monoid are equivalent to so-called weakly ordered monoids. We provide an alternative algorithm, based only on the simple observation that an R-trivial monoid may be realized by upper triangular matrices. This approach is inspired by results in the field of coupled cell network dynamical systems, where L-trivial monoids (the opposite notion) correspond to so-called feed-forward networks. We first show that our algorithm works for ZM, after which we prove that it also works for RM where R is an arbitrary ring with a known complete system of primitive orthogonal idempotents. In particular, our algorithm works if R is any field. In this respect our result constitutes a considerable generalization of the results in Berg et al. [J. Algebra 348 (2011) 446–461]. Moreover, the system of idempotents for RM is obtained from the one our algorithm yields for ZM in a straightforward manner. In other words, for any finite R-trivial monoid M our algorithm only has to be performed for ZM, after which a system of idempotents follows for any ring with a given system of idempotents.}},
  author       = {{Nijholt, Eddie and Rink, Bob and Schwenker, Sören}},
  issn         = {{0219-4988}},
  journal      = {{Journal of Algebra and Its Applications}},
  keywords     = {{Applied Mathematics, Algebra and Number Theory}},
  number       = {{12}},
  publisher    = {{World Scientific Pub Co Pte Ltd}},
  title        = {{{A new algorithm for computing idempotents of ℛ-trivial monoids}}},
  doi          = {{10.1142/s0219498821502273}},
  volume       = {{20}},
  year         = {{2020}},
}