ABSTRACT
Next, we prove that every nonzero ideal I of F[x; u] has the form I = F[x; u)x"'. From Theorem 1.3.1 we know that I = F[x; u]g where g is a polynomial of minimal degree k in I, and that g may be replaced by a polynomial f E I of degree k and leading coefficient 1. Now if
then
k-1 xf-fx = E (u(a;) - a;)xi+1
has fewer terms than f. Hence by Theorem 1.3.1 we conclude that xf-fx = 0, and so a; E Ftr for each i = 0, 1, ... , k-1. Moreover, for every c E F we see that u"'(c)f-fc E F[x; u] has degree less than k, so it must be 0, implying
Let X be a set of finitely or infinitely many symbols {indeterminates). The polynomial ring A[X] with commuting indeterminates over A is defined as the set of all finite sums of finite products of powers of the indeterminates x; E X with coefficients from A, and the addition and multiplication are the usual ones. We can speak also of the ring of polynomials A(X) with noncommuting indeterminates. Here we mean that the indeterminates commute with the elements of A but not with each other.
