Assignment:
Let x and y belong to a commutative ring R with prime characteristic p.
a) Show that (x + y)^p = x^p + y^p
b) Show that, for all positive integers n, (x + y)^p^n = x^p^n + y^p^n.
c) Find elements x and y in a ring of characteristic 4 such that (x + y)^4 != x^4 + y^4.
Provide complete and step by step solution for the question and show calculations and use formulas.