MA 3362 Lecture 09 - Some Basic Properties of Rings (cont.)
Friday, September 26, 2008.
Objectives: Continue proving basic ring properties
Back to the ring properties.
Unities are unique. A ring does not have to have a unity, but if it does, then it can only have one.
The proof is similar to the one proving that the zero is unique, but we should be careful not to assume
commutativity for products, since multiplication does not have to be commutative.
Theorem 1. A ring can have at most one unity.
This is a non-existence proof, so we’ll do a proof by contradiction. “At most one . . .” proofs are pretty
common, so it’s worth remembering how to start. The conceivable possibilities here are that there are no
unities, one unity, two unities, three unities, etc. We want to eliminate the two or more cases. In all of the
two or more cases, if true, we would be able to find two distinct unities. Showing that this is impossible
leaves only the zero or one cases. OK?
Let R be a ring. Suppose that there are two, distinct unities, 1 and 10 . In particular,
(1)
1·a =a
(2)
(3)
(4)
a·1 =a
10 · a = a
a · 10 = a,
and
remembering that · is not necessarily commutative, and
(5)
1 6= 10 .
The trick here is to consider 1 · 10 , and since both are unities, we have
(6)
1 · 10 = 1and
1 · 10 = 10 .
By the uniqueness of · (and I suppose transitivity), it must be that
(7)
1 = 10 .
This contradicts the “distinctness” of the two unities.
Multiplicative inverses are unique. We defined multiplicative inverses specifically for fields, which are
commutative. In a ring with unity, which need not be commutative, a pair of multiplicative inverses a and
b will be required to satisfy
(8)
ab = ba = 1.
Uniqueness for multiplicative inverses is a bit more interesting than it might seem. In the proof of the
uniqueness of additive inverses, we used the additive cancellation law. The proof of the cancellation law used
the existence of additive inverses. In a ring with unity, some elements may have multiplicative inverses, and
others might not. As a result, we have to be more careful in this proof. Of course, this proof would work
for the additive case.
Theorem 2. Every element of a ring has at most one multiplicative inverse.
Again, this is an “at most one . . . ” proof, so we’ll assume that there are at least two distinct inverses, and
then find a contradiction.
1
MA 3362 Lecture 09 - Some Basic Properties of Rings (cont.)
2
Let R be a ring with unity, and a ∈ R. Suppose that a has two distinct multiplicative inverses, b and c. In
particular, we have
(9)
(10)
(11)
ab = 1
ba = 1
ac = 1
(12)
(13)
ca = 1
b 6= c.
and
The trick is to use associativity on the expression bac, which tells us that we can interpret this expression as
(ba)c or b(ac), and we would get the same result each time. By the associative property, therefore, we have
(14)
(15)
bac = (ba)c = 1c = c
bac = b(ac) = b1 = b,
and therefore,
(16)
c = b.
This contradicts the “distinctness” of c and b.
Now that we know that multiplicative inverses are unique (if they exist), we will use the following notation.
Notational Convention 1. In a ring R, if a ∈ R has a multiplicative inverse, we will call it a−1 . Be
careful to note that a−1 may not exist.
Multiplying by 0 gives you 0 - didn’t we already know that? All we require in a ring for multiplication
is the associative property and the distributive property, so we might get some pretty weird multiplications.
Proving this establishes the fact that the ring properties are sufficient for the 0 to act as we expect.
Theorem 3. Let R be a ring, 0 the zero in R, and a ∈ R. It must be true that a · 0 = 0 · a = 0.
Let R be a ring, and let a ∈ R. By the definition of a zero,
(17)
a + 0 = a.
By the uniqueness of products, we can multiply both of these expressions by a (both on the same side), and
they will still be equal.
(18)
a(a + 0) = aa.
By the distributive property,
(19)
aa + a0 = aa.
The additive inverse of the element aa is −(aa), and
(20)
−(aa) + aa = 0.
Therefore, we can add −(aa) to both sides of equation (19), and still maintain equality to get
(21)
−(aa) + aa + a0 = −(aa) + aa,
and this simplifies to
(22)
0 + a0 = 0,
and
(23)
a0 = 0.
MA 3362 Lecture 09 - Some Basic Properties of Rings (cont.)
3
Working with negative sign notation. We are using the notation −a to denote the additive inverse of
a. It means nothing more than that. On the other hand, that little line behaves as we’re used to.
Notational Convention 2. Basically,
(24)
(−a) + a = a + (−a) = 0
looks like it would, if a were a real number, although that can be far from the case. Symbolically, however,
we might as well just write
(25)
−a + a = a − a = 0.
With real numbers, we casually write things like (−x)y = −xy, with the expectation that xy − xy = 0.
Whatever this might mean in an arbitrary ring, the symbols will look the same.
Let R be a ring, and let a, b ∈ R. Both of these elements have additive inverses, −a and −b. Let’s multiply
them together in different combinations. First, a · (−b). We can get a clue about what this equals by adding
it to ab.
(26)
ab + a(−b) = a(b + (−b)) = a · 0 = 0,
by the distributive property. We see that a(−b) must be an additive inverse of ab, and since inverses are
unique, this must be the additive inverse, −(ab). Similarly,
(27)
ab + (−a)b = (a − a)b = 0b = 0.
Apparently,
(28)
−(ab) = (−a)b = a(−b).
Since these are all the same, there is no problem with using the following notation.
Notational Convention 3. We will use the notation −ab interchangeably with
(29)
−ab = −(ab) = (−a)b = a(−b).
Look at what the distributive property looks like.
(30)
a(b − c) = a(b + (−c)) = ab + a(−c) = ab − ac
It looks just like the real number symbols.