Topics in algebra Chapter IV: Commutative rings and modules I - 1

(a) R is a local ring.
(b) The set of non-units of R is contained in a proper ideal of R.
(c) The set of non-units of R forms an ideal.
Proof. Let T be the set of all non-units of R.
(a) ⇒ (b) Let P be the unique maximal ideal of R and a /∈ P . If a is not
a unit, then (a) ⊆ M for some maximal ideal of R, so that a ∈ M = P , a
contradiction. Therefore a is a unit and T ⊆ P .
(b) ⇒ (c) Suppose that T ⊆ P for some proper ideal P . As a proper ideal,
P consists of non-units, therefore P ⊆ T and then T = P is an ideal of R.
(c) ⇒ (a) By assumption, T is an ideal of R. Let I be an ideal of R such
that I T ; then there is an element a ∈ I \ T , so that a is a unit and
I = R. Therefore every proper ideal of R is contained in T and T is the
unique maximal ideal of R.
Example 3.18 Let R be a ring and S = R − ∪ n i=1Pi, where Pi is a prime
ideal of R for every i; then RS is a semi-local ring and the maximal ideals of
RS is contained in {P e 1 , . . . , P e n}. If S = R − P for some prime P of R, then
RS (often written as RP ) is a local ring with maximal ideal PS = P RP .
The ring RP is called the localization of R at P . There is a well-known
local ring: the formal power series ring.
Definition 3.19 Let F be a field; then the set
∞
F [[x]] = {
i=0
aix i | ai ∈ F }
of all formal power series is a ring under the formulas:
and
where ck =
i+j=k
∞
aix i +
i=0
∞
bix i =
i=0
∞
(ai + bi)x i
i=0
∞
( aix i ∞
) · ( bix i ) =
i=0
i=0
∞
ckx k ,
k=0
aibj = a0bk + a1bk−1 + · · · + akb0.
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