Corollary

Tag 04F1, permanent. Cite it as [Bourbaki, Tag 04F1].

Commutative Algebra, chapter III, § 2. General results on filtered rings and modules

Let $\mathbf{A}$ be a semi-local ring (Chapter II, § 3, no. 5), $m_i$ its distinct maximal ideals $(1 \leq i \leq q)$ and

r = m_1 \cap m_2 \cap \cdots \cap m_q

its Jacobson radical. The Hausdorff completion $\hat{\mathbf{A}}$ of $\mathbf{A}$ with respect to the r-adic topology is a semi-local ring, canonically isomorphic to the product $\prod_{i=1}^q \hat{\mathbf{A}}_{m_i}$ , where $\hat{\mathbf{A}}_{m_i}$ is the Hausdorff completion ring of the local ring $\mathbf{A}_{m_i}$ with respect to the $(m_i \mathbf{A}_{m_i})$ -adic topology.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Exercise 4, § 3
Remark, § 4
Exercise 8, § 2

Cites

Chapter II, § 3, no. 5