Show that there cannot be an integral domain with 6 elements.

Let \(R\) be a commutative ring with characteristic \(p\), where \(p\) is a prime. Show that \((a+b)^{p^n}=a^{p^n}+b^{p^n}\) and \((a-b)^{p^n}=a^{p^n}-b^{p^n}\) for all integers \(n \ge0\) . Also show that if \(R\) is an integral domain, then the mapping \(a \mapsto a^p\) is a monomorphism of \(R\).

Show that every endomorphism of a field is either trivial or a monomorphism. Hence show that every non-trivial endomorphism of a finite field is an automorphism.

Let \(I\) and \(J\) be non-zero ideals of an integral domain. Prove that \(I \cap J \neq \{0\}\) .

Show that the ring \(M_n(\mathbb{F})\) of \(n \times n\) matrices over a field \(\mathbb{F}\) is a simple ring. Hence deduce that simple rings may have one-sided proper ideals.

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 \(R\).