Definition 1

Let $A$ be a local ring. An $A$-algebra $B$ is said to be an elementary enlargement of $A$ if $B$ is isomorphic to the $A$-algebra $A[X]$, or else if there exists a monic polynomial $P$ in $A[X]$, whose image in $\kappa_A[X]$ is irreducible, such that $B$ is isomorphic to the $A$-algebra $A[X]/(P)$.

Let $B$ be an elementary enlargement of $A$. From the preceding there result the following properties:

a) The ring $B$ is local and the canonical homomorphism from $A$ into $B$ is local and flat, and in particular injective (I, § 3, no 5, prop. 8).

b) The residue field $\kappa_B$ of $B$ is a monogenic extension of the residue field $\kappa_A$ of $A$. If $\kappa_A$ is of finite degree $d$ over $\kappa_A$, then $B$ is a free $A$-module of rank $d$.

c) We have $m_B = m_A B$. In particular, if $A$ is a field, the same is true of $B$. An extension of fields is an elementary enlargement if and only if it is monogenic.

d) If $A$ is noetherian, the same is true of $B$.