Let $R$ be a ring in which $a^2=a, \forall a \in R$ (such a ring is called a Boolean ring). Show that $2a=0, \forall a\in R$ , and $R$ is commutative. Further show that the only Boolean ring that is an integral domain is $\mathbb{Z_2}$.

Let $R$ be a ring and $a,b \in R$such that $ab=ba$ . Prove that for any positive integer  $(a+b)^n =a^n +\left( \begin{array}{c} n \\ 1 \end{array} \right)a^{n-1}b+ \cdots +\left( \begin{array}{c} n \\ n-1 \end{array} \right) ab^{n-1}+b^n$ .

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.