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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.
#340153 on Dec 2023