Page 150 - DMTH403_ABSTRACT_ALGEBRA

Unit 14: Rings

We define ‘+’ and ‘.’ in Z + iZ to be the usual addition and multiplication of complex numbers. Notes
Thus, foram + in and s + it in Z + iZ,
(m + in) + (s + it) = (m + s) + i(n + t), and
(m + in) . (s + it) = (p – nt) + i(mt + ns).
Verify that Z + iZ is a ring under this addition and multiplication. (This ring is called the ring of
Gaussian integers, after the mathematician Carl Friedrich Gauss.)
Solution: Check that (Z + iZ, +) is a subgroup of (C, I–). Thus, the axioms RI-R4 are satisfied. You
can also check that
((a + ib) . (c + id)) . (m + in) = (a + ib) . ((c + id) . (m + in))

 a + ib, c + id, m + in  Z + iZ.
This shows that R5 is also satisfied.
Finally, you can check that the right distributive law holds, i.e.,
((a + ib) + (c + id)) . (m + in) = (a + ib) . (m + in) + (c + id) . (m + in) for any a + ib, c + id, m +
in  Z + iZ.
Similarly, you can check that the left distributive law holds. Thus, (Z + iZ, + , .) is a ring. The
operations that we consider in it are not the usual addition and multiplication.

Example: Let X be a non-empty set, (XI ) be the collection of all subsets of X and A
denote the symmetric difference operation. Show that ((X), A, n) is a ring.
Solution: For any two subsets A and B of X,
A  B = (A\B)  (B\A)

It is clear that ( (X), A) is an abelian group. You also know that  is associative. Now let us see
if  distributes over A.
Let A, B, C E  (X). Then
A  (B  C) = A  [(B\C)  (C\B)]
= [A  (B\C)][A  (C\ B)], since n distributes over U.

= [(A  B)\(A  C)][(A  C)\(A  B)], since  distributes over complementation.
= (A  B) A (A  C).
So, the left distributive law holds.
Also, (B  C)  A = A  (B  C), since  is commutative.
= (A  B) P (A  C)
= (B  A) A ( C  A).
Therefore, the right distributive law holds also.
Therefore, (  (X), A, ) is a ring.
So’ far you have seen examples of rings in which both the operations defined on the ring have
been commutative. This is not so in the next example.

Example: Consider the set
 a   11 a 12   
M (R) =  a a  a, a , a , and a; are real numbers 
12
21
2
   21 22   

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