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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.


8. Let A be a ring #= 0. Show that the set of prime ideals of A has minimal ele-

ments with respect to inclusion. 


7. Let A be a ring in which every element x satisfies xn = x for some n > 1 

(depending on x). Show that every prime ideal in A is maximal. 


6. A ring A is such that every ideal not contained in the nilradical contains a non-

zero idempotent (that is, an element e such that e2 = e #= 0). Prove that the 

nilradical and Jacobson radical of A are equal. 


4. In the ring A[x], the Jacobson radical is equal to the nilradical.


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.


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.


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