CHAPTER 1

Differential Ideal s

Basic convention s

In thi s section , w e fix som e basic conventions an d terminolog y whic h will be in

force throughou t th e book .

We will alway s understand a "ring " t o b e a "commutativ e ring" .

A differential ring R i s a ring with a specified derivation DR (w e could be technical

and sa y a differential rin g is an ordere d pair

(R,DR)).

Th e derivation, whic h will be

abbreviated D whe n n o confusio n arises , is a map fro m R t o R satisfyin g

D(a +b) = D{a) + D{b) an d D(ab) = aD{b) + D(a)b fo r a,b £ R.

Sometimes w e will denot e D(x) b y x' an d D n{x) b y x^ n\

Every rin g i s a differentia l rin g usin g th e zer o derivation ; non-trivia l example s

will appea r shortly .

The usua l formula e fo r derivative s hol d i n differentia l rings :

2(1)=0,

D{xn)=nxn~lD{x)

fo r n 1,

D{x/y) = ' — fo r y a unit .

y

A differentia l rin g which i s a field wil l b e calle d a differential field.

Throughout, we will let F b e a differential field of characteristic zero.

An extension o f differential ring s is an inclusion R C S o f differential ring s suc h

that D

s

(a) = DR{a) fo r al l a e R. A differential rin g extension o f F i s a differentia l

F-algebra.

The se t o f constants o f a differentia l rin g R i s the kerne l o f DR. I t i s a subrin g

of R (eve n a differential subrin g with the zero derivation). I f R i s a differential field,

its constant s ar e a subfield .

A homomorphism o f differential ring s is a rin g homomorphis m / : R — S suc h

that D

s

(f(a)) = f{D

R

{a)) fo r al l a G R.

We are now going to consider som e of the standard construction s i n ring theor y

and not e wha t happen s i n th e differentia l context .

First w e look a t quotien t rings : le t R b e a differential ring . A differential ideal

is a n idea l I of R suc h tha t D{I) C / . I t i s eas y t o se e that a n idea l generate d b y

a subse t X satisfyin g D(X) C l i s a differentia l ideal . I f / i s a differentia l ideal , i t

follows that the quotient ring R/I i s a differential rin g with DR/f(a+I) = D R{a)+I.

DRJJ

i s well define d sinc e D preserve s / .

l

http://dx.doi.org/10.1090/ulect/007/01