Homework nr.3 Ideal

              Theorem of ideals   

  J. F. Rittf introduced the idea of irreducible system of algebraic differential
equations and showed that every system of such equations is equivalent
to a finite set of irreducible systems.
      One of the objects of this paper is to develop a special type of abstract
ideal theory which has Ritt's theorem as a consequence. The elements of our
ideals are polynomials in unknowns yi, • • • , yn and a certain number of their
derivatives. Following Ritt, we call these polynomials forms. The coefficients
in these forms are assumed to be elements of a differential field fJ of characteristic
zero.f A differential field is a commutative field (as in abstract algebra)
whose elements a, b, ....  have unique derivatives ai, bi, ..... which are elements
of the field. These derivatives must satisfy the rules (a+b)i = ai+bi
and (ab)i =aib+abi. The totality of these forms with coefficients in F
differential ring R. We consider differential ideals, which are ideals containing
together with any element its derivative. An example given by Ritt
shows that there exists a differential ideal of R having no finite subset, such
that every element of the ideal is a linear combination of elements of the
subset and their derivatives with forms of R as coefficients.**

        Certain results of Ritt suggested that we consider, as our purpose permits,
only differential ideals which have the property that if they contain an element a of R, they contain any element b of R such that a positive power of b is a.  We call these differential ideals perfect differential ideals.

    We show that every perfect differential ideal of R is the intersection of a finite number of prime perfect differential ideals.

         The use of perfect differential ideals was suggested by the foUowing two
results of Ritt:
        (a) Every infinite system of forms has a finite subsystem whose manifold of
solutions is identical with that of the infinite system*
       (b) Let F1, . . . ,Fr;  G be forms such that G has every solution of the system
F1, . . . , Fr. Then some power of G is a linear combination of the F i and a certain
number of their derivatives with forms for coefficients.
       We obtain abstract theorems that specialize to a combination of these
results of Ritt. For instance, we show that every perfect differential ideal of R
has a finite subset such that every form of the ideal has a power which is a linear
combination of the forms of the subset and their derivatives with forms of R for
coefficients. The proof of this basis theorem is like the proof of Ritt's result
(a) in fundamental respects, but there are essential differences. We also obtain
an abstract generalization of Ritt's result (b). The conciseness of the
proof of this theorem is an indication of the simplicity of our theory.
        Having established the basis theorem, the development of our ideal theory
follows approximately the well known methods of E. Noether.

                                      Perfect differential ideals
    1. We consider a fixed differential ring R of characteristic zero.
    The intersection of any arbitrary set of differential ideals is a differential
ideal. For let a be any element of the intersection. Then a is an element of
every ideal of the set; hence the derivative ax is in the intersection. The intersection,
which is known to be an ideal, is then a differential ideal. The intersection
of any arbitrary set of perfect differential ideals is a perfect differential
ideal. Let a and b be elements of R such that a is in the intersection
and some power of b is a. Then a is in every ideal of the set, hence also b.
Therefore the intersection is a perfect differential ideal.

         Let a be an arbitrary set of elements of R. We notice that R is a perfect
differential ideal. The intersection of the differential ideals containing a will
be called the differential ideal [a] determined by a. [a] is uniquely defined.

       The intersection of all perfect differential ideals containing a we call the
perfect differential ideal {a} determined by a. {a} is uniquely defined.



      Let a be any set of elements of R. We shall denote by a' the set consisting
of all elements of R which have a positive integral power in a. Using the set a
of the preceding paragraph, we define a recursively as follows:
                                                   a1 = {a},
                                               an = {a n - 1 }                                   (n = 2, 3, 4, . . . )
Let ß denote the totality of elements of the sets an. Then ß is a perfect differential
ideal and is contained in { a }, hence is {a}. This means that any element t of {a} is in some an with a sufficiently large subscript.