Assignment:
Q1. Show that a non-zero ring R is local (i.e. it has a unique maximal ideal) if and only if x or 1 - x is a unit, where x is a member of R.
Q2. Show why or why not are the following four rings local: {a/b belongs to Q s.t. b is not divisible by a prime p}, Z[x]/(X^3), F[[x]] (where F[[x]] is the ring formed by the set of all formal power series p(x) = a_0 + (a_1)(x) + (a_2)(x^2) + ... where a_i belongs to the field F), {p(x)/q(x) belongs to Q(x) s.t. q(0) = 0}.
Q3. For the previous rings that you decided are not local, slightly change their definition (without proof) such that you get a local ring.
Provide complete and step by step solution for the question and show calculations and use formulas.