If $a$ is a nilpotent element in a ring $R$ with identity 1, then show that  $1-a$ is a unit in $R$. Further, show that if $R$ is commutative, then $1-ab$ is a unit for all  $b \in R$ .

Let $R$ be a non-zero commutative ring with identity. If $R$ has no non-trivial ideals, prove that $R$ is a field.

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 $n$.