Let be two nontrivial rings linked by an application ($ K\stackrel{\vartheta }{\mapsto } L $) having the following properties: $ \text{(i)}\quad x,y\in K\implies \vartheta (x+y) = \vartheta (x) +\vartheta (y) $ $ \text{(ii)}\quad \vartheta (1)=1 $ $ \text{(iii)}\quad \vartheta \left( x^3\right) =\vartheta^3 (x) $ [b]a)[/b] Show that if $ \text{char} (L)\ge 4, $ and $ K,L $ are fields, then $ \vartheta $ is an homomorphism. [b]b)[/b] Prove that if $ K $ is a noncommutative division ring, then it’s possible that $ \vartheta $ is not an homomorphism.

Let $ X $ be the power set of set of $ \{ 0\}\cup\mathbb{N} , $ and let be a function $ d:X^2\longrightarrow\mathbb{R} $ defined as $$ d(U,V)=\sum_{n\in\mathbb{N}}\frac{\chi_U (n) +\chi_V (n) -2\chi_{U\cap V} (n)}{2} , $$ where $ \chi_W (n)=\left\{ \begin{matrix} 1,& n\in W\\ 0,& n\not\in W \end{matrix} \right. ,\quad\forall W\in X,\forall n\in\mathbb{N} . $ [b]a)[/b] Prove that there exists an unique $ V' $ such that $ \lim_{k\to\infty} d\left( \{ k+i|i\in\mathbb{N}\} , V'\right) =0. $ [b]b)[/b] Demonstrate that for all $ V\in X $ there exists a $ v\in\mathbb{N} $ with $ d\left( \left\{ \frac{3}{2} -\frac{1}{2}(-1)^{v} \right\} , V \right) >\frac{1}{k} . $ [b]c)[/b] Let $ f: X\longrightarrow X,\quad f(X)=\left\{ 1+x|x\in X\right\} . $ Calculate $ d\left( f(A),f(B) \right) $ in terms of $ d(A,B) $ and prove that $ f $ admits an unique fixed point.