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.

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