2 1. DIFFERENTIA L IDEAL S

Next w e consider ring s o f fractions: le t R b e a differentia l rin g an d let g b e

a multiplicativel y close d subse t o f R containin g 1 and not containing 0 . Defin e

D :

Q-lR-

Q'

lR

b y

(

. qD{a) ~aD(q)

D(a/q) = ^ •

To check that D is well defined, i t is convenient to first recall the connection betwee n

derivations an d dual numbers .

DEFINITION.

Le t R be a ring. Th e ring of dual numbers over R is the ring R[s] —

R + Re wher e e

2

= 0.

Since e is nilpotent, a n element x = a + be of R[e] is a unit i f and onl y if a is

a uni t o f R.

It i s easy t o check tha t a n additive ma p D : JR— R is a derivation o f i? i f and

only if aD = (id R,D) : i£— i?[e] is a ring homomorphism .

Now let R be a differential rin g as above. The n w e have a ring homomorphis m

aD : R -^ R[e] and we consider it s composition b with th e homomorphism R[s] —

g~li?[£].

Th e resulting map sends ^ € g t o x = #/l +D{q)/\e. Sinc e #/l is a unit

of Q~ lR, x i s a unit o f g -1/?[£]. Thi s mean s tha t b extends t o a homomorphis m

c : Q~ lR - » g ~ */£[£], an d it is clear tha t c ha s the form (id,E), wher e £ i s a

derivation extendin g D. The n i t follows fro m th e observations abov e tha t

as desired .

Special case s o f thi s constructio n includ e th e fact tha t th e fraction fiel d o f a

differential integra l domai n i s a differentia l field , an d in a uniqu e way ; if R i s a

differential integra l domain , E it s fraction field , an d / a n element o f E no t in R,

then R[l/f] C E i s a differential subring .

Next, le t R b e a differentia l rin g an d let S an d T b e differential i^-algebras .

Define

D:S®RT -+S® RT b y D( s 0 0 - D s(s) 0 f + s 0 D T(t).

It is easy to verify tha t this formula give s a derivation provided it is well defined .

For th e latter, first w e assume tha t D

R

= 0 . The n D

s

an d DT are .R-linear, s o

D = Ds 0 id

T

+ ids ® Z)r i s well defined an d is a derivation o f S 0/? T. I n general,

let C denot e th e ring o f constants o f R. The n b y the preceding remar k D i s well

defined o n S gc T an d it preserves th e setJlr = { r ® l — lg)r} , an d hence passe s

to th e quotient b y the ideal generate d b y X.

Finally, w e consider th e adjunction o f indeterminates. Le t R b e a differentia l

ring. Th e ring of differential polynomials over R in the variable Y is the polynomial

ring R{Y} = R[{Y^^\i — 0.1.2....}] i n the countable numbe r o f indeterminate s

F (,) , wit h the derivation defined to extend that of R and such that D(F ( / ) ) = 7 ( / + l ) .

If R is a differential integra l domai n wit h fractio n field F , then R{ Y} i s also a

domain, whos e fractio n field wil l be denoted F(Y).

Elements o f R{Y} ca n be regarded a s differential operator s o n R: fo r we have

an obvious ring homomorphism R{ Y} — End(,R) which sends Y^ t o Dl an d a e R

to lef t multiplicatio n b y a. W e will typically confuse th e differential polynomia l and

the corresponding operator. I n particular, we have the homogeneous linear differential