Abstract Algebra Answers

3.3. If in a ring R every x ∈ R satisfies x² = x, prove that R

must be commutative

(A ring in which x² = x for all elements is called a Boolean ring).

3.1. If R is a ring and a, b, c, d ∈ R, evaluate (a + b)(c + d).

3.2. Prove that if a, b ∈ R, then (a + b)² = a² + ab + ba + b² where

by x² we mean xx.

1. Let x be a nilpotent element of a ring A. Show that 1 + x is a unit of A. Deduce 

that the sum of a nilpotent element and a unit is a unit. 

44. Prove that if a and b are different integers, then there exist infinitely 

many positive integers n such that a+n and b+n are relatively prime. 

Let K be a field and f : Z → K the homomorphism of

integers into K.

a) Show that the kernel of f is a prime ideal. If f is an embedding,

then we say that K has characteristic zero.

b) If kerf f= {0}, show that kerf is generated by a prime number

p. In this case we say that K has characteristic p.