Nilradical and radical ideals

If $A$ is a commutative ring and $A[[x]]$ is the ring of formal power series over $A$, show that if $f = \sum_{n=0}^\infty a_n x^n$ is nilpotent, then all the elements $a_i \in A$ are nilpotent. If $A$ is Noetherian, prove the converse, i.e., that if all the elements $a_i$ are nilpotent, then $f$ is nilpotent.

Show that the nilpotent elements of a commutative ring form a subring.

Show that the nilpotent elements of a commutative ring form a subring.

Show that $Z_n$ has a nonzero nilpotent element if and only if $n$ is divisible by the square of some prime.

Let $R$ be a commutative ring and let $A$ be any ideal of $R$. Show that the nil radical of $A$, $N(A) = \left\{ r \in R \mid r^n \in A \text{ for some positive integer } n\ (n \text{ depends on } r)\right\}$, is an ideal of $R$.

(Definitions used elsewhere in this chapter's exercises: a prime ideal $A$ of a commutative ring $R$ is a proper ideal of $R$ such that $a, b \in R$ and $ab \in A$ imply $a \in A$ or $b \in A$; a maximal ideal of $R$ is a proper ideal $A$ such that, whenever $B$ is an ideal of $R$ with $A \subseteq B \subseteq R$, then $B = A$ or $B = R$.)

Let $R = Z_{27}$. Find

Let $R = Z_{36}$. Find

Let $R$ be a commutative ring. Show that $R/N( \left\langle 0\right\rangle )$ has no nonzero nilpotent elements.

Let $A$ be an ideal of a commutative ring. Prove that $N(N(A)) = N(A)$.