Exercise 4, § 10

A ring A is called pseudoregular if for every element $a$ of A, there exist an integer $n$ and an element $x$ of A such that $a^nxa^n=a^n$. An absolutely flat ring (VIII, p. 171, Exercise 27) is pseudoregular.

a) Prove that a pseudoregular ring is a Zorn ring (VIII, p. 171, Exercise 26) in which every noninvertible element is a left and right zero divisor.

b) Prove that every quotient ring of a pseudoregular ring is pseudoregular (cf. VIII, p. 171, Exercise 27, d)).

c) Prove that the center Z of a pseudoregular ring A is pseudoregular (prove that if $a\in Z$ and $x\in A$ satisfy $a^nxa^n=a^n$, then $a^{2n}x^k$ belongs to Z for every $k\geqslant 1$).

d) Let $p$ be a prime number, $U_p$ the $p$-primary component of the group $\mathbf{Q}/\mathbf{Z}$, and A the (commutative) algebra of the group $U_p$ over the field $\mathbf{F}_p$. Prove that the ring A is pseudoregular but $A^{\mathbf{N}}$ is not.