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Sorry in advance if this is too silly. Let $R$ be a right semihereditary ring and $P$ a projective right $R$-module. It is well-known that finitely generated (thus projective) submodules of $P$ form a lattice. Let now $F$ be a finitely generated submodule of $P$ and $S$ a (possibly infinitely generated) projective submodule of $P$. Is $F+S \subseteq P$ also necessarily projective? I'm mainly interested in the case of $R$ being noetherian, when $F \cap S$ is f.g. and thus projective (so in this case we know at least that $F+S$ has projective dimension $\leq 1$).

I would suspect that this is false, but I'm unable to create a counterexample. Thanks for any input.

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    $\begingroup$ You say you're mainly interested in the case where $R$ is noetherian. But a noetherian semihereditary ring is hereditary, so $F+S$ is projective because every submodule of $P$ is projective. $\endgroup$ Jan 4, 2014 at 11:07
  • $\begingroup$ I see that my in-advance apology has not come in vain, this was indeed silly. Thank you for the comment. However, I would still be interested in the non-noetherian setting. $\endgroup$
    – Fred.Fred
    Jan 4, 2014 at 11:56
  • $\begingroup$ What is an example of a semi hereditary non-hereditary ring? $\endgroup$
    – Will Sawin
    Jan 7, 2014 at 18:17
  • $\begingroup$ For example an infinite product of fields. $\endgroup$
    – Fred.Fred
    Jan 8, 2014 at 12:12

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