Why Read This Book? - Index of

9.3 Ring Properties 297
Theorem 9.3.7 If R is a ring with unity, then the unity element is unique.
Even though elements of a ring with unity are not assumed to have multiplicative
inverses, some of them might. If for a given x there exists y such that
xy = yx = e, then x is called a unit of the ring. Notice that Exercise 9.3.1 implies
that zero is not a unit.
EXERCISE 9.3.8 Find, with verification, all units in the following rings.
(a) Z
(b) Z12
(c) Z7
(d) D2×2 from Exercise 9.2.5
(e) The near ring of functions from Exercise 9.1.7
(f) The power set ring from Exercise 9.1.8
Divisibility in a ring is defined in pretty much the same way it is in the integers,
except that we must distinguish between left and right divisors.
Definition 9.3.9 Let R be a ring, and let a and b be elements of R, where a is
nonzero. Then a is called a left divisor of b provided there exists nonzero k ∈ R
such that ak = b. Similarly, a is called a right divisor of b if there exists nonzero
k ∈ R such that ka = b. IfR is a commutative ring and there exists nonzero k ∈ R
such that ak = b, we say simply that a is a divisor of b, or that a divides b, and we
write a | b.
If a divides b, it does not necessarily mean that the k such that ak = b or ka = b
is unique. In a more specialized ring we will study in Section 9.8, however, we will
have uniqueness.
EXERCISE 9.3.10 Find nonzero elements a, b, k1,k2 in each of the following
rings where ak1 = b and ak2 = b, but k1 is different from k2.
(a) Z12
(b) Z2×2
Having defined divisors, we can now define what it means for an element of
a ring to be prime, though we will not really look into any of its properties until
Section 9.8. To motivate the term by returning to the positive integers, a prime
p has exactly two distinct positive integer divisors, 1 and p. Thus by definition,
1 is not prime. So if p is a prime number and p = ab is any factorization of p
into positive integers a and b, then either a = 1orb = 1. In the ring of integers,
primes are extended to include the negatives of the prime natural numbers. Thus
in the integers, for p to be prime means that if p = ab is any factorization of p into