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
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 :
fo r n 1,
D{x/y) = ' fo r y a unit .
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
(a) = DR{a) fo r al l a e R. A differential rin g extension o f F i s a differentia l
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
(f(a)) = f{D
{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.
i s well define d sinc e D preserve s / .
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