Exercise 26, § 9

A ring A is called a Zorn ring if every left ideal of A that is not a nil ideal contains a nonzero idempotent.

a) Prove that in a Zorn ring, every right ideal that is not a nil ideal contains a nonzero idempotent (observe that for every nonnilpotent element $a$ of A, there exists an $x\in A$ such that $xaxa=xa\not= 0$).

b) Prove that the radical of a Zorn ring is a nil ideal and that a Zorn ring without zero divisors is a field.

c) Prove that a primitive ring with nonzero socle is a Zorn ring (cf. VIII, p. 129, Exercise 4, b) and p. 130, Exercise 7, a)). Give an example of a primitive ring that is not a Zorn ring (cf. VIII, p. 132, Exercise 13) and an example of a primitive ring whose socle is nonzero but whose center is not a Zorn ring (use the method of Exercise 12).

d) Let K be a commutative field and A the subring of $K(X)^{\mathbf{N}}$ consisting of the sequences $s= (f_n)$ for which there exists a polynomial $\varphi (s)\in K[X]$ such that $f_n$ is equal to $\varphi (s)$ for $n$ sufficiently large. Prove that A is a commutative Zorn ring, without radical, containing elements that are neither invertible nor zero divisors, but that its image $\varphi (A)$ is not a Zorn ring.