Exercise 12, § 12

Let $A_1$ be the K-algebra $K(X)[U]_{1,d/dX}$ (VIII, p. 11), $A_2$ the K-algebra K[[T]], and A the K-algebra $A_1\otimes_KA_2$. Identify $A_1$ and $A_2$ with subalgebras of A.

a) Prove that $A_2$ is the center of A, that A does not contain any zero divisors, and that the invertible elements of A are those of $A_2$.

b) Prove that the two-sided ideals of A are the ideals AT$^n$ for $n\geqslant 0$.

c) Prove that the ring A is primitive even though its center has nonzero radical (observe that a maximal left ideal of A containing $1-$ TU does not contain any two-sided ideals other than 0).