Page 45 - DMTH502_LINEAR_ALGEBRA

Unit 1: Vector Space over Fields

(v) Addition is commutative. Notes
R (R, .) is a semigroup, i.e.,
2
(vi) r r = r (mod m) R
1 2
r being the remainder obtained after dividing r r by m if r r m.
1 2 1 2
(vii) (r r ) r r (r r )  , , ,r r r R i.e., multiplication is associative. R Distributive axiom
1 2 3 1 2 3 1 2 3 3
is satisfied, i.e.,
(viii) r (r + r ) = r r + r r and (r + r ) r = r r + r r for r r , r R.
1 2 3 1 2 1 3 2 3 1 2 1 3 1 1 2 3
Hence (R, +, .) is a ring.

Special Types of Rings

Some special types of rings are discussed below:

1. Commutative Rings: A ring R is said to be a commutative, if the multiplication composition
in R is commutative, i.e.,
ab = ba a, b R.

2. Rings with Unit Element: A ring R is said to be a ring with unit element if R has a
multiplicative identity, i.e., if there exists an element R denoted by 1, such that
1 . a = a . 1 = a a R.

The ring of all n × n matrices with elements as integers (rational, real or complex numbers)
is a ring with unity. The unity matrix

1 0 0  0
0 1 0  0
I n 0 0 1  0
0 0 0  1

is the unity element of the ring.
3. Rings with or without Zero Divisors: While dealing with an arbitrary ring R, we may find
elements a and b in R neither of which is zero, and their product may be zero. We call such
elements divisors of zero or zero divisors.

Definition: A ring element a ( 0) is called a divisor of zero if there exists an element b ( 0)
in the ring such that either
ab = 0 or ba = 0

We also say that a ring R is without zero divisors if the product of no. two non-zero elements of
same is zero, i.e., if
ab = 0 either a = 0 or b = 0 or both a = 0 and b = 0.

Cancellation Laws in a Ring

We say that cancellation laws hold in a ring R if

ab = ac (a 0) b = c
and ba = ca (a 0) b = c where a, b, c, are in R.
Thus in a ring with zero divisors, it is impossible to define a cancellation law.

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