DHARM
96 MATHEMATICAL FOUNDATION OF COMPUTER SCIENCE
(iv) If ring X contains more than two elements then there exist distinct elements 0, x and
y. Since x + y ≠ 0, if x + y = 0 then x = y which contradicts the facts that x and y are
distinct.
Now consider, x • y, if x • y = 0 then x and y are zero divisors.
Conversely, if x • y ≠ 0 then x y (x + y) = x y x + x y^2 [Distributive law]
⇒ x x y + x y^2 [∴ x y = y x]
⇒ x y + x y [∴ x^2 = x & y^2 = y]
⇒ 0 [from (ii)]
Therefore, x y and x + y are zero divisors.
Example 4.16. Show that a Boolean ring with more than two elements has zero divisors.
Sol. Let a ring B in which every element is idempotent (∴ x^2 = x for any x ∈ B) is called a
Boolean ring. A Boolean ring is commutative and also x + x = 0 for any x ∈ B.
Let B contain more than two distinct elements these are 0, a, and b. Consider a + b and a b
a + b ≠ 0 for a + b = 0 ⇒ a = b
If a b = 0, then a and b are zero divisors and if a b ≠ 0, then a b (a + b) = a b a + a b^2
⇒ a(b a) + a b
⇒ a(a b) + a b
⇒ a^2 b + a b
⇒ a b + a b = 0
Hence a b and a + b are zero divisors.
4.13 Fields..................................................................................................................................
An algebraic system (X, +, •), where + and • are two usual binary addition and multiplication
operations, is called a field if,
- (X, +, •) is a commutative ring with unity and
- Every nonzero element of X is invertible.
For example, let X is the set of real numbers (R) then (R, +, •) is a field. Consider
another examples, assume X is the set of all complex numbers (C) or set of rational numbers
(Q) then algebraic structure (C, +, •) and (Q, +, •) are fields. But, if X = I or set of integers then
algebraic structure (I, +, •) of integers is not a field, because (I, +, •) is a commutative ring and
a ring with unity but, not every nonzero element of I is invertible.
Example 4.17. Show that algebraic system (Y, + 5 , × 5 ) is a field, where Y = {0, 1, 2, 3, 4} and
the operations + 5 and + 5 are addition modulo 5 and multiplication modulo 5 respectively.
Sol. Since, algebraic system (Y, + 5 , × 5 ) is a ring. Now we will show that this is a commutative
ring with unity and every nonzero element of Y is invertible. To prove that we construct the
operation table for the operation × 5 that is shown below (Fig. 4.7)
× 5 0 1 234
000000
101234
202413
303142
404321
Fig. 4.7 Operation table for X 5
