Page 46 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes Theorem 11: A ring has no divisor of zero if and only if the cancellation law s holds in R.
Proof: Suppose that R has no zero divisors. Let a, b, c, be any three elements of R such that a 0,
ab = ac.

Now, ab = ac ab – ac = 0
a (b – c) = 0
b – c = 0 (because R is without zero divisors and a 0)
b = c.

Thus the left cancellation law holds in R. Similarly, it can be shown that right cancellation law
also holds in R.
Conversely, suppose that the cancellation laws hold in R.

Let a, b R and if possible let ab = 0 with a 0, b 0 then ab = a . 0 (because a . 0 = 0)
Since a 0, ab = a . 0 b = 0 (by left cancellation law)
Hence we get a contradiction to our assumption that b 0 and therefore the theorem is established.

Division Ring

A ring is called a division ring if its non-zero elements form a group under multiplication.
Pseudo ring: A non-empty set R with binary operations ‘+’ and ‘.’ satisfying all the postulates of
a ring except right and left distributive laws, is called a pseudo ring if

(a b ) (c d ) a c a d b c b d for all , , ,d R
c
a
b
Subrings
Definition: Let R be a ring. A non-empty subset S of the set R is said to be a subring of R if S is closed under
addition and multiplication in R and S itself is a ring, for those operations.

If R is any ring, then {0} and R are always subrings of R. These are said to be improper subrings.
The subrings of R other than these two, if any, are said to be proper subrings of R.
Evidently, if S is a subring of a ring R, it is a sub group of the additive group R.

Theorem 12: The necessary and sufficient condition for a non-empty subset S of a ring R to be a
subring of R are
(i) a, b S a – b S,

(ii) a, b S ab S.
Proof: To prove that the conditions are necessary let us suppose that S is a subring of R.
Obviously S is a group with respect to addition, therefore,
b S – b S
Since S is closed under addition

a S, b S a S, – b S a + (– b) S
a – b S.

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