Ring theory

Introduction to rings: defining a ring and giving examples. Knowledge of sets, proofs, and mathematical groups are recommended. Practice problems: 1.) Which of the following sets are rings? If it is, prove it. If not, say which property of rings fails for that set (there may be more than one). a) N = {1,2,3,4,5,...} under normal addition and multiplication b) A = {a+b*sqrt(2) | a,b are rationals} under normal addition and multiplication c) B = {all polynomials p(x) with integer coefficients} d) C = {all polynomials p(x) with integer coefficients, where deg( p(x) ) is even} 2.) The set {0,2,4,6,8,10,12} is a ring with unity under the operations of addition mod 14 and multiplication mod 14. What is the unity of this ring? 3.) Let R be a commutative ring with unity, and let U(R) denote the set of units (elements with multiplicative inverses) of R. Prove that U(R) is a group under the multiplication defined in R.