Example 2

Let A be a local ring, and I a set of indices. Denote by $A[(X_i)_{i \in I}]$ the local ring of the polynomial ring $A[(X_i)_{i \in I}]$ at the prime ideal $m_A A[(X_i)_{i \in I}]$. The A-algebra $A[(X_i)_{i \in I}]$ is an inflation of A. In fact, furnish the set I with a well-ordering; let $\Lambda$ be the well-ordered set obtained by adjoining to I a greatest element $\omega$. For $i \in I$, identify $A[(X_j)_{j < i}]$ with a subalgebra $B_i$ of $B = A[(X_i)_{i \in I}]$, and put $B_\omega = B$. The family $(B_\lambda)_{\lambda \in \Lambda}$ satisfies conditions a), b), c).