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
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