CHAPTER (2)
RINGS
2.1Definition: A non-empty set R, together with two binary compositions + and . is said to form a Ring, if
(i) a+(b+c) = (a+b)+c for all a,b,c in R.
(ii) a+b = b+a for all a,b in R.
(iii) ( some element o (called zero) in R such that
a+o = a = o+a for all a in R.
(iv) for each a in R, ( an element (-a) in R such that
a+(-a) = o = (-a)+a
(v) a.(b.c) = (a.b).c for all a,b,c in R.
(vi) Left distributive law:
a.(b+c) = a.b+a.c
Right distributive law:
(b+c).a = b.a+c.a
for all a,b,c in R.
2.1.1EXAMPLES OF RINGS:
Example (1): Let Z be the set of integers. Then
Z[i] = {a+ib: a,b ( Z}
forms a ring under usual addition and multiplication of complex numbers.
Example (2): Let us consider the set X={0,1,2,3,4,5}. Then X is a ring under addition and multiplication modulo 6.
Example (3): Let G be an additive abelian group with at least two elements. We define a binary composition on the elements of G as follows:
For any x,y in G, x.y=o .
Then < G , + , . > is a ring.
Example (4): The set R consisting of a single element o with two binary operations defined by 0+0=0 and 0.0=0 is a ring. This ring is called the null ring or the zero ring.
Example (5): The set M of all n(n matrices with their elements as real numbers is a ring with respect to addition and multiplication of matrices as the two ring compositions.
Example (6): Let X be a non-empty set. Then P(X), the power set of X forms a ring under + and . defined by
A+B = (A(B)((A(B)...