Definition 2

Let $A$ be a local ring. An $A$-algebra $B$ is said to be an enlargement of $A$ if there exists a well-ordered set $\Lambda$ having a greatest element $\omega$, and an increasing family $(B_\lambda)_{\lambda \in \Lambda}$ of subalgebras of $B$ satisfying the following conditions:

a) We have $B_\omega = B$ and the ring $B_\lambda$ is local for every $\lambda \in \Lambda$.

b) If $\alpha$ is the smallest element of $\Lambda$, the $A$-algebra $B_\alpha$ is isomorphic to $A$.

c) Let $\nu \neq \alpha$ in $\Lambda$ and let $S_\nu$ be the set of $\lambda \in \Lambda$ such that $\lambda < \nu$. If $S_\nu$ has no greatest element, one has $B_\nu = \bigcup_{\lambda \in S_\nu} B_\lambda$; if $S_\nu$ has a greatest element $\mu$, then $B_\nu$ is an elementary inflation of $B_\mu$.

Let $B$ be a ring and $\rho : A \to B$ a ring homomorphism. One says that $\rho$ is an inflation (resp. an elementary inflation) if the $A$-algebra defined by $\rho$ has this property. If this is so, $\rho$ is injective.