Exercise 7, § 9

Let A be a ring, and let I be the set of ordinals $\alpha$ such that Card($\alpha$ )$\leqslant$ Card(A) (Set Theory, III, §6, p. 241, Exercise 10). Define two-sided ideals $\mathfrak{N}_{\alpha}$ for all $\alpha \in I$ as follows by transfinite induction. Take $\mathfrak{N}_0=\mathfrak{N}$ (Exercise 6, b)). If $\alpha$ admits a successor $\alpha + 1$, then $\mathfrak{N}_{\alpha+1}$ is the two-sided ideal such that $\mathfrak{N}_{\alpha+1}/\mathfrak{N}_{\alpha}$ is the sum of the nilpotent two-sided ideals of $A/\mathfrak{N}_{\alpha}$. If $\alpha$ is a limit ordinal, take $\mathfrak{N}_{\alpha}$ to be the union of the $\mathfrak{N}_{\beta}$ for $\beta < \alpha$. Let $\tau$ be the smallest ordinal such that $\mathfrak{N}_{\tau+1}=\mathfrak{N}_{\tau}$; the ideal $\mathfrak{P}=\mathfrak{N}_{\tau}$ is called the prime radical of A.

a) Prove that we have $\mathfrak{N}\subset \mathfrak{P}\subset \mathfrak{G}$ and that $\mathfrak{P}$ is the smallest semiprime nil ideal of A.[^1]

b) Prove that $\mathfrak{P}$ is the intersection of the semiprime two-sided ideals of A (cf. VIII, p. 149, Exercise 3).

c) Prove that $\mathfrak{P}$ is the intersection of the prime two-sided ideals of A (loc. cit., c)). d) Prove that $\mathfrak{P}$ is the set of elements $a$ of A with the following property: every sequence $(a_n)_{n\geqslant 0}$ of elements of A such that $a_0=a$ and $a_{n+1}\in a_nAa_n$ for every $n$ has finite support (same method as in loc. cit.).

[^1]: For an example of a ring such that $\mathfrak{P}\not=\mathfrak{G}$, see R. Baer, Radical ideals, Amer. J. of Math. 65 (1943), p. 540.