Mathematical Foundation of Computer Science

DHARM

ALGEBRAIC STRUCTURES 97

Form the operation table our observations is follows,
l It has an identity element 1, so it is a ring with unity.
l Rows are similar to the corresponding columns, i.e. row 1 = column 1, row 2 = col-
umn 2, and others. Hence, ring is commutative.
l Every nonzero element of Y is invertible, i.e.
Let 1–1 is y so, 1 × 5 y = 1(identity) ⇒ y = 1, so 1–1 = 1.
Similarly, 2–1 will be 3 [∴ 2× 5 3 = 1], 3–1 will be 2 [∴ 3× 5 2 = 1], 4–1 will be 4 [∴ 4× 5
4 = 1]. Therefore, (Y, + 5 , × 5 ) is a field.
This ring is an integral domain because it is a commutative ring with unity. Also it is a
ring without zero divisors, due to entries of the first column of the operation table which
shows that for all x ∈ Y, x × 5 y = 0 when y = 0 and whatever the x is.
Example 4.18. Show that the algebraic system (Y, + 6 , × 6 ) is not a field where Y = {0,1, 2, 3, 4,
5}.
Sol. Since we have already saw that algebraic system (Y, + 6 , × 6 ) is a ring (example 1.30). To
show that (Y, + 6 , × 6 ) is a field then it should be a commutative ring with unity followed that
every nonzero element of Y is invertible. Observe the operation table for × 6.

× 6 0 1 2345

0000000

1012345

2024024

3030303

4042042

5054321

l Operation table posses an identity element 1 hence it is a ring with unity.

l Row1 = column 1, row 2 = column2, and so on hence ring is commutative.

l Every nonzero element is not invertible, for example 2, 3, and 4 are not invertible. It

means for no y ∈ Y, we have 2 × 6 y = 1, 3 × 6 y = 1, and 4 × 6 y = 1.

Therefore, (Y, + 6 , × 6 ) is not a field.

Skew Field

Let (X, +, •) be a field then it is called a skew field if it a ring with unity and where every

nonzero element has multiplicative inverse. Hence, if a skew field is commutative then it

becomes a field.

For example, let M be a set of matrices i.e.,

M =

uiv

wix

wix

uiv

+

−+

+

−

F

HG

I

KJ

where u, v, w, and x ∈ R and the binary operations are usual addition (+) and multiplication

(•) then (M, +, •) is a ring. It is a ring with unity, where unity matrix of M is,

MI =

10

01

10 0 0

00 1 0

F

HG

I

KJ=

++

−+ −

F

HG

I

KJ

ii

ii

and every nonzero element has a multiplicative inverse, since M–1 exists if M is singular

matrix or | M | ≠ 0. Since, | M | = u^2 + v^2 + w^2 + x^2 ≠ 0. Therefore, (M, +, •) is a skew field.