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 \(n .\)