Let $R$ be a commutative ring. Show that the set $N=\{ a\in R |a^m =0$ for some positive integer m$\}$ is an ideal of $R$ which is contained in every prime ideal of $R$. Further show that $R/N$ is a ring with no non-zero nilpotent elements. $(N$ is called the nilradical of $R)$.

Let $R$ be a commutative ring and $I$ an ideal of $R$, and let $\sqrt{I} =\{a \in R |a^m \in I$ for some positive integer $m\}$ Show that

(i) $\sqrt{I}$ is an ideal of $R$ and $I \subseteq \sqrt{I}$ ,

(ii) $\sqrt{\sqrt{I}}=\sqrt{I}$ ,

(iii) If $R$ is with identity and $\sqrt{I} =R$, then $I=R$ . What is $\sqrt{I}$, when $I=\{0\}$. ($\sqrt{I}$ is called the radical of $I$).

If $n > 1$ is a positive integer and $n=p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}$ is the prime factorization of $n$, then show that  $\sqrt{\langle n \rangle }=\langle p_1p_2 \cdots p_k \rangle$ .

Let $R$ be a commutative ring and $I$ an ideal of $R$. Let Jac $I$ denote the intersection of all maximal ideals of $R$ that contain $I$.

(i) Prove that Jac $I$ is an ideal of $R$, and $\sqrt{I} \subseteq$ Jac $I$.

(ii) If $n >1$ is an integer, describe $n\mathbb{Z}$ in terms of the prime factorization of $n$.

Let $R$ be a commutative ring and $I,J$ be ideals of $R$. Prove that if $I+J =R$, then $IJ = I \cap J$ .

Let $R=C([0,1])$ be the ring of continuous real-valued functions on the interval $[0,1]$. Let $T \subseteq [0,1]$, and let

   $I(T) = \{f \in R : f(x) =0 \forall x \in T \}$ .

(i) Prove that $I(T)$ is an ideal of $R$.

(ii) If $x \in [0,1]$ and $m_x =I(\{x\})$, show that $R/M_x \cong \mathbb{R}$, and hence $M_x$ is a maximal ideal of $R$.

A subset $S$ of a ring $R$ is said to be multiplicative if $ab \in S \;\forall \; a,b \in S$. Show that $P$ is a prime ideal of a commutative ring $R$ if and only if $R-P$ is multiplicative.

Let $I$ be the principal ideal $\langle x \rangle$ of the ring $\mathbb{Z}[x]$. Prove that $I$ is prime but not maximal.

Let $R$ be a commutative ring with identity in which the set of all non-units forms an ideal $M$. Show that $M$ is a maximal ideal of $R$ and it contains every proper ideal of $R$. Also show that for each $r \in R$ either $r$ or $1-r$ is a unit.

Let $R$ be a commutative ring. Show that $a|b \Leftrightarrow Ra \supset Rb, \forall a,b \in R$. Further if $R$ contains the identity, show that for any $a \in R$, if $Ra$ is maximal, then $a$ is an irreducible element of $R$.

Let $R$ be a non-trivial ring with identity in which every subring is an ideal. Show that $R \cong \mathbb{Z}$ or $\mathbb{Z_n}$ for some $n .$