nLab
Pierce spectrum

Let $R$ be a commutative ring. By $E(R)$ denote the Boolean algebra of idempotents of $R$ whose meet operation is given by the multiplication of $R$. the Pierce spectrum $\mathrm{Idl}(E(R))$ of $R$ is the poset (in fact locale) of ideals of $E(R)$. There is a sheaf $\overline{R}$ of indecomposable rings (rings whose only idempotents are $0$ and $1$) on $\mathrm{Idl}(E(R))$, called the Pierce sheaf, whose ring of sections over the principal ideal $(e)$ (where $e$ is a prime filter in $E(R)$) is $R_e$.

F. Borceux, G. Janelidze, Galois Theories, Cambridge studies in advanced mathematics, 72, Cambridge University Press 2001.

P. T. Johnstone, Stone Spaces, Cambridge studies in advanced mathematics 3, Cambrdge Univ. Press 1982.