Exercise 12, § 7

Let A be a ring and $d$ a derivation of A. Denote the ring $A[D]_{1,d}$ (VIII, p. 11) simply by A[D]. Every element $P\not= 0$ of A[D] can be written uniquely as $\sum^m_{k=0}a_kD^k$ with $a_k\in A$ and $a_m\not= 0$; we call $m$ the degree of P and $a_m$ its leading coefficient.

a) Let $\mathfrak{b}$ be a two-sided ideal of A[D], and let $n$ be the minimum of the degrees of the nonzero elements of $\mathfrak{b}$. Prove that the set consisting of 0 and the leading coefficients of the elements of $\mathfrak{b}$ of degree $n$ is a two-sided ideal of A.

b) Suppose that the ring A is quasi-simple, that A is a $\mathbf{Q}$-algebra, and that the derivation $d$ is not an inner derivation (III, §10, No. 6, p. 560). Prove that the ring A[D] is quasi-simple.

c) Suppose that A is a commutative field of characteristic zero and that $d\not= 0$. Prove that every (left or right) ideal of A[D] is monogenous (consider an element of minimal degree in such an ideal). Deduce that A[D] is left and right Noetherian, quasi-simple, and without zero divisor but does not contain any minimal right or left ideals.