Let $R$ be a commutative semihereditary ring (i.e. every finitely generated ideal of $R$ is projective https://en.wikipedia.org/wiki/Hereditary_ring). Then is $R$ a finite direct product of Prufer domains ( https://en.wikipedia.org/wiki/Pr%C3%BCfer_domain) ?
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No. Consider, for example, the von Neumann regular ring $R=\prod_{\aleph_0}\mathbb{Q}$. A finitely generated one-sided ideal in a von Neumann regular ring is generated by an idempotent, hence it is projective. (It is also easy to check this fact directly for $R$.) In particular, $R$ is semihereditary. On the other hand, $R$ contains an infinite system of pairwise orthogonal idempotents and therefore cannot be a product of finitely many domains.
