In theorem 14.7 in the book below, they claim that the singular submodule of $R$ cannot contain a direct summand of $R$, but I cannot understand why.
2026-04-09 16:59:18.1775753958
Can $Z(R_R)$ contain a direct summand of $R$?
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A direct summand of $R_R$ takes the form of $eR$ where $e$ is an idempotent element.
If $eR$ were contained in the right singular submodule of $R$ for a nontrivial idempotent $e$, that would imply the right annihilator $\mathscr{r}(e)$ is an essential submodule of $R$.
But it isn't: it is easy to prove that $\mathscr{r}(e)=(1-e)R$, which is a nontrivial summand, and therefore impossible to be essential.