Augmented rings. By a ring we understand a commutative associative unital
algebra1 over a basic field , that is, the word “ring” shall mean a ring in the
sense of Chapter 1 in [AM69] which is simultaneously a -vector space and whose
structure operations are compatible with the scalar multiplication, i.e.
α(r + r ) = αr + αr , and α(r r ) = (αr )r = r (αr ),
for α ∈ , r , r ∈ R.
Definition 1.1. Let R be a ring with unit e and ω : → R the morphism
given by ω(1) := e. A morphism : R → is an augmentation of R if ω = or,
✑ ✑ ✸
A ring with an augmentation is an augmented ring.
The subspace R := Ker is called the augmentation ideal of R. Since the
quotient R/R is isomorphic to the field , the augmentation ideal is always maximal.
In this way, each augmentation determines a maximal ideal in R. Vice versa, each
maximal ideal m ⊂ R defines an augmentation R → R/m over the field R/m which
however, as we will see in Example 1.3 below, need not be isomorphic to the basic
Example 1.2. The unital ring [[t]] of formal power series with coeﬃcients in
is augmented, with the augmentation : [[t]] → given by
It turns out that [[t]] is a local Noetherian ring, with the unique maximal ideal (t)
and residue field , see [AM69, Chapter 1] for the terminology.
Example 1.3. Every α ∈ determines an augmentation
: [t] → of the
polynomial ring [t] given by
(f) := f(α), for f ∈ [t]. On the other hand, given
an augmentation : [t] → , take α := (t). It is clear that, for this α, = α.
There is therefore a one-to-one correspondence between augmentations of [t] and
points in the aﬃne plane .
The augmentation ideal of
: [t] → is the maximal ideal generated by
(t − α). If is algebraically closed, then this assignment is one-to-one, i.e. there
Definition 1.15 below.