Page 155 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes (iv) For a, b, c  R,
a(b - c) = a(b + (– c))
= ab + a(– c), by distributivity.

= ab + (– (ac)), from (ii) above.
= ab – ac.
If k is an integer (k  2) such that the sum of k elements in a ring R is defined, we define the sum
of (k + 1) elements a , a ..., a in R, taken in that order, as a + .., + a k+1 = (a + ..... + a ) + a .
k
k+l
k
1
l 2
k+1
In the same way if k is a positive integer such that the product of k elements in R is defined, we
define the product of (k + 1) elements a , a , ..., a (taken in that order) as
1
2
k+l
a .a ... a = (a .a .... .a ) . a .
l
12
k+1
1
2
k+1
k
As we did for groups, we can obtain laws of indices in the case of rings also with respect to both
+ and ., in fact, we have the following results for any ring R.
(i) If m and n are positive integers and a  R, then
a . a = a m+n , and
m
n
(a )” = a .
mn
m
(ii) If m and n are arbitrary integers and a, b  R, then
(n + m)a = na + ma,
(nm)a = n(ma) = m(na),
n(a + b) = na + nb,
m(ab) = (ma)b = a(mb), and
(ma) (nb) = mn (ab) = (mna)b.
(iii) If a + a , ..., a,, b , ..., b ,  R then
n
1
2
1
(a , + ... + a ) ( b + ... + b )
1
n
m
1
= a b + ... + a b + a a + ... + a b + ... + a b + ... + a b .
2 1
m n
m 1
2 n
1 n
l
l
There are several other properties of rings that we will be discussing throughout this block. For
now let us look closely at two types of rings, which are classified according to the behaviour of
the multiplication defined on them.
14.3 Two Types of Rings
The definition of a ring guarantees that the binary operation multiplication is associative and,
along with +, satisfies the distributive laws. Nothing more is said about the properties of
multiplication. If we place restrictions on this operation we get several types of rings. Let us
introduce you to two of them now.
Definition: We say that a ring (R, +, .) is commutative if . is commutative, i.e., if ab = ba for all a,
b  R.
For example, Z, Q and R are commutative rings.
Definition: We say that a ring (R, +, .) is a ring with identity (or with unity) if R has an identity
element with respect to multiplication, i.e., if there exists an element e in R such that
ae = ea = a for all a  R.

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