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.

2.10. Let H be the subgroup generated by two elements a, b of a group G. Prove that if ab = ba, then H is an abelian group.

2.9. Let a and b be integers.

(a) Prove that the subset aZ + bZ = {ak + bl | l, k ∈ Z } is a subgroup of Z.

(b) Prove that a and b + 7a generate the subgroup aZ + bZ.

2.8. Let a, b be elements of a group G. Assume that a has order 5 and a³b = ba³. Prove that ab = ba.

2.7. If G is a group such that (ab)2 = a2b2 for all a, b ∈ G, then show that G must be abelian.

2.6. If G is a group in which (ab)ⁱ = aⁱbⁱ for three consecutive integers i for all a, b ∈ G, show that G is abelian.