Let $d$ be any integer. Prove that $\mathbb{Z}[\sqrt{d}] =\{a+b\sqrt{d} \; |a.b \in \mathbb{Z} \}$ is an integral domain and $\mathbb{Z}[\sqrt{d}] =\{a+b\sqrt{d} \; |a.b \in \mathbb{Z} \}$ is a field.

Show that any ring with 6 elements is commutative.

If a non-zero subring $S$ of a ring $R$ has an identity $1^\prime$ but $R$ has either no identity or the identity of $R$ is different from $1^\prime$ , then show that $1^\prime$ is a zero-divisor in $R$ .

Prove that all the non-zero elements in an integral domain have same additive order, which is the characteristic of $R$ if char $R >0$and infinite if char $R=0$ .

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.