nLab Pierce spectrum

Idea

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 $\bar{R}$ 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 $R_e$.

Related concepts

• spectrum (geometry)

References

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.