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Let $M$ be a right $R$-module ($R$ has unity). Recall that $M$ is called square-free if $M$ does not contain two nonzero isomorphic submodules with zero intersection. $M$ is called CS if every submodule is essential in a summand. $M$ is called C3 if the sum of any two summands with zero intersection is also a summand. I seek a module with the following requirements.

  1. $M$ is square-free,
  2. $M$ is CS.
  3. $M$ is NOT C3.
  4. $M$ is NOT nonsingular.

I also need the full details (if possible). Thanks in advance.

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  • $\begingroup$ In the title you say NOT square free. Is the title correct or the body? $\endgroup$
    – Kevin Casto
    Commented Dec 3, 2023 at 19:08
  • $\begingroup$ Oh, sorry. I edited the title. Thanks. $\endgroup$
    – Hussein Eid
    Commented Dec 3, 2023 at 19:11
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    $\begingroup$ I don't know anything about this, but for this sort of questions, it is generally appreciated if you give some motivation as to why you are looking for this particular set of conditions; also, you should probably say what you already know, like, do you have examples satisfying any 3 of the 4 conditions? $\endgroup$
    – Gro-Tsen
    Commented Dec 3, 2023 at 21:30

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The only example I know of a square-free module that does not have the (C3) property occurs as Example 6.1 in a paper I wrote with Ryszard Mazurek and Michał Ziembowski titled "Commuting idempotents, square-free modules, and the exchange property." I believe that module is also not nonsingular. I'm not sure if it is CS. You might look at the literature citing that paper, to see if some more recent work addresses your question directly.

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    $\begingroup$ I'm doing research on module theory and I'm very interested in your great work. I have this paper by the way. It's wonderful. In fact, I'm looking for a pseudo-continuous module which is NOT quasi-continuous. Indeed, the module can't be nonsingular because Yasser and Yousif in their paper $C4$-modules proved that any nonsingular pseudo-continuous module is quasi-continuous. $\endgroup$
    – Hussein Eid
    Commented Dec 4, 2023 at 18:37
  • $\begingroup$ I appreciate if you can help me in proving or disproving the module is CS. $\endgroup$
    – Hussein Eid
    Commented Dec 4, 2023 at 20:02

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