Page 156 - DMTH403_ABSTRACT_ALGEBRA

Unit 14: Rings

Can you think of such a ring? Aren’t Z, Q and R examples of a ring with identity? Notes
Definition: We say that a ring (R, +, .) is a commutative ring with unity, if it is a commutative
ring and has the multiplicative identity element 1.
Thus, the rings Z, Q, Rand C are all commutative rings with unity. The integer 1 is the
multiplicative identity in all these rings.
We can also find commutative rings which are not rings with identity. For example, 2Z, the ring
of all even integers is commutative. But it has no multiplicative identity.

Similarly, we can find rings with identity which are not commutative. For example, M (R) has
2
the unit element   1 0  .
 0 1 
But it is not commutative. For instance,

if A =   1 0  and B    0 1  , then
 2 0   0 2 

AB =   1 0  0 1     0 1  and
 
 2 0   0 2   0 2 

BA =   0 1  1 0     2 0  and
 
 0 2   2 0   4 0 
Thus, AB  BA.
Now, can the trivial ring be a ring with identity? Since 0 . 0 = 0, 0 is also the multiplicative
identity for this ring. So (( 0 ), +, .) is a ring with identity in which the additive and identities
coincide. But, if R is not the trivial ring we have the following result.
Theorem 2: Let R be a ring with identity 1. If R  { 0 } then the elements 0 and 1 are distinct.

Proof: Since R  { 0 } ,  a  R, a  0. Now suppose 0 = 1. Then a = a . 1 = a . 0 = 0 (by Theorem 1).
That is, a = 0, a contradiction. Thus, our supposition is wrong. That is, 0  1.
Now let us go back when will A × B be commutative? A × B is commutative if and only if both
the rings A and B are commutative. Let us see why. For convenience we will denote the operations
in all three rings A, B and A × B by + and . . Let (a, h) and (a’, b’)  A × B.

Then (a, b) . (a’, b’) = (a’, b’) . (a, b)
( a.a ‘, b . b’) = (a’. a, b’ . b)
a.a’ = a’.a and b . b’= b’. b .
Thus, A × B is commutative iff both A and B are commutative rings.

We can similarly show that A × B is with unity iff A and B are with unity. If A and B have
identities e and e respectively, then the identity of A × B is (e , e ).
2
1
2
1
Now we will give an important example of a non-commutative ring with identity. This is the
ring of real quaternions. It was first described by the Irish mathematician William Rowan
Hamilton (1805-1865). It plays an important role in geometry, number theory and the study of
mechanics.

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