Page 48 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes = ab + [–(ac)]
= ab – ac.
Also, (b – c) a = [b + (–c)] a
= ba + (–c) a (right distributive law)

= ba + [– (ca)] = ba – ca.

Example 18: Suppose M is a ring of all 2 × 2 matrices with their elements as integers, the
addition and multiplication of matrices being the two ring compositions. Then M is a ring with
left zero-divisor.

0 0
Solution: The null matrix O = is the zero element of ring M.
0 0

1 0 0 0
A = and B = are two non-zero elements of M.
0 0 1 0

1 0 0 0 0 0
Now AB = = 0.
0 0 0 1 0 0
Hence M is a ring with left zero divisor.

Example 19: Prove that the ring of integers is a ring without zero divisors.
Solution: Since the product of two non-zero integers is never zero, it is the ring without zero
divisors.

Example 20: Prove that the ring of residue classes modulo a composite integer m possess
proper zero divisors.
Solution: Let m = ab i.e., a and b are two factors of m.
Then ab 0 (mod m)

But a (mod m) and b 0 (mod m).
Hence the residue classes {a} and {b} are proper zero-divisors.

Example 21: Prove that the totality R of all ordered pairs (a, b) of real numbers is a ring
with zero divisors under the addition and multiplication defined as

(a, b) + (c, d) = (a + c, b + d),
(a, b) (c, d) = (ac, bd), (a, b), (c, d) R.
Solution: First of all, we prove that R is ring. We have
(R ) : [(a, b) + (c, d)] = (a + c, b + d) R
1
Hence R is closed for addition.
(R ) : [(a, b) + (c, d)] + (e, f) = (a + c, b + d) + (e, f)
1
= ((a c ) e , (b d ) ) f

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