DHARM94 MATHEMATICAL FOUNDATION OF COMPUTER SCIENCE
l There exists precisely one element – x, i.e.
x + (– x) = 0 for all x ∈ X, or
instead of y + (– x) we simply write y – x for any x, y ∈ X.
l For a natural number n (1, 2, 3......) we put
n • x = x + x + ....... + x (n times)
l For a negative number n′ (– 1, – 2, .......) we put
n′ • x = (– n) • x = n • (– x) = (– x) + (– x) + ......... + (– x) (n times)
l For the number 0, we put, for any x ∈ X
0 • x = x (the zero – element of X)
l Let I is the set of all integers (0, ± 1, ± 2,........, ± n,......) then for all p, q ∈ I and x, y
∈ X, we have
(p + q) • x = p • x + q • x ; p • (q • x) = (p q) x ;
and q(x + y) = q x + q y
l Therefore, it is easy to verify that,
x • (– y) = – (x y) ; (– x) • y = – (x y) ; (– x) (– y) = x • y ;
also, x • (y – z) = x • y – x • z ;(y – z) • x = y • x – z • x
also, x • (y 1 + y 2 + ...... + yn) = x y 1 + x y 2 + ........ + x yn
also, (y 1 + y 2 + ...... + yn) • x = y 1 x + y 2 x + ........ + yn x
further we have,
xp • xq = xp+q = xq • xp ;
also, (xp)q = xpq ;
l If (X, +, •) is a commutative ring, i.e., x • y = y • x for all x, y ∈ X then
(x • y)p = xp • yp ;Ring with zero divisors
Let (X, +, •) be a ring. If x, y ∈ X such that x • y = 0 (Additive identity) for x ≠ 0 and y ≠ 0, then
ring is zero divisor ring and the element x is called zero divisor of ring.
Consider the ring (Y, + 6 , × 6 ) that was discussed in previous example, this is a zero
divisor ring. Because, searching of two elements x, y ∈ Y i.e., x × 6 y should be zero provided x
≠ 0 and y ≠ 0. If we take two elements 2 and 3 from set Y then (2 × 6 3) = 0 (additive identity),
another pair of elements is 4 and 3 i.e., (4 × 6 3) = 0.
Ring without zero divisor
Let (X, +, •) be a ring. If x, y ∈ X such that x. y = 0 (Additive identity) for either x = 0 or y = 0,
then (X, +, •) is a ring without zero divisor.
For example, (I, +, •) is a ring without zero divisor. Because, for any pair of elements x,
y ∈ I, x ∈ y = 0 only when either element x = 0 or y = 0, or both are zero.
Integral Domain
A ring (X, +, •) is called an integral domain if,
l It is a commutative ring,
l It is a ring with unity, and
l It is a ring without zero divisors.