Exercise 50, § 1

Let A be a pre-$\lambda$-ring. By a $\lambda$-ideal of A is meant an ideal $a$ of A such that $\lambda_n(a) \subset a$ for all $n \in J$.

a) Let $a$ be a $\lambda$-ideal of A. Now prove that there exists a unique pre-$\lambda$-ring structure $\lambda : A/a \to \Lambda(A/a)$ on $A/a$ such that the canonical projection $A \to A/a$ is a $\lambda$-morphism.

b) The $\lambda$-ideals of A are precisely the kernels of the $\lambda$-morphisms of A into other pre-$\lambda$-rings.

c) Let $(a_i)_{i \in I}$ be a family of $\lambda$-ideals of A. Then $\bigcap a_i$ and $\sum a_i$ are $\lambda$-ideals of A.

d) Let A be a $\lambda$-ring. If $a$ and $a'$ are $\lambda$-ideals of A, then $aa'$ is a $\lambda$-ideal.

e) Let A be a $\lambda$-ring and let $a \in A$. The ideal $a$ of A generated by $a, \lambda_2(a), \lambda_3(a), ...$ is a $\lambda$-ideal.